Sequences

392,541 sequences

Expansion of 1/((1-2x)(1-6x)(1-8x)(1-9x)).A026561
Expansion of 1/((1-2x)(1-6x)(1-8x)(1-9x)).
Expansion of 1/((1-2*x)*(1-6*x)*(1-8*x)*(1-10*x)).A026562
Expansion of 1/((1-2*x)*(1-6*x)*(1-8*x)*(1-10*x)).
a(n) = T(n, floor(n/2)), where T is given by A026552.A026563
a(n) = T(n, floor(n/2)), where T is given by A026552.
a(n) = Sum_{j=0..n} T(n, j), where T is given by A026552.A026564
a(n) = Sum_{j=0..n} T(n, j), where T is given by A026552.
a(n) = 6*a(n-2), starting with 1, 3, 9.A026565
a(n) = 6*a(n-2), starting with 1, 3, 9.
a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026552.A026566
a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026552.
a(n) = Sum_{i=0..2*n} Sum_{j=0..i} T(i, j), where T is given by A026552.A026567
a(n) = Sum_{i=0..2*n} Sum_{j=0..i} T(i, j), where T is given by A026552.
Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = [ (i+1)/2 ] for i >= 1; and for i >= 2 and 2 <=j <= i - 2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i + j is even, T(i,j) = T(i-1,j-2) + T(i-1,j) if i + j is odd.A026568
Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = [ (i+1)/2 ] for i >= 1; and for i >= 2 and 2 <=j <= i - 2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i + j is even, T(i,j) = T(i-1,j-2) + T(i-1,j) if i + j is odd.
a(n) = T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.A026569
a(n) = T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.
a(n) = A026568(n,n-1), also a(n) = number of integer strings s(0),...,s(n) counted by A026568 such that s(n)=1.A026570
a(n) = A026568(n,n-1), also a(n) = number of integer strings s(0),...,s(n) counted by A026568 such that s(n)=1.
a(n) = T(n,n-2), T given by A026568. Also a(n) = number of integer strings s(0), ..., s(n) counted by T, such that s(n) = 2.A026571
a(n) = T(n,n-2), T given by A026568. Also a(n) = number of integer strings s(0), ..., s(n) counted by T, such that s(n) = 2.
a(n) = T(n,n-3), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=3.A026572
a(n) = T(n,n-3), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=3.
a(n)=T(n,n-4), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.A026573
a(n)=T(n,n-4), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.
T(2n,n), T given by A026568.A026574
T(2n,n), T given by A026568.
T(2n,n-1), T given by A026568.A026575
T(2n,n-1), T given by A026568.
T(2n,n-2), T given by A026568.A026576
T(2n,n-2), T given by A026568.
T(2n-1,n-1), T given by A026568.A026577
T(2n-1,n-1), T given by A026568.
T(2n-1,n-2), T given by A026568.A026578
T(2n-1,n-2), T given by A026568.
T(n,[ n/2 ]), T given by A026568.A026579
T(n,[ n/2 ]), T given by A026568.
T(n,0) + T(n,1) + ... + T(n,n), T given by A026568.A026580
T(n,0) + T(n,1) + ... + T(n,n), T given by A026568.
Expansion of (1 + 2*x) / (1 - x - 4*x^2).A026581
Expansion of (1 + 2*x) / (1 - x - 4*x^2).
a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026568.A026582
a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026568.
a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=2n, T given by A026568.A026583
a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=2n, T given by A026568.
Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.A026584
Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.
a(n) = T(n,n), T given by A026584. Also a(n) is the number of integer strings s(0), ..., s(n) counted by T, such that s(n)=0.A026585
a(n) = T(n,n), T given by A026584. Also a(n) is the number of integer strings s(0), ..., s(n) counted by T, such that s(n)=0.
Number of labeled essential directed acyclic graphs (DAGs).A026586
Number of labeled essential directed acyclic graphs (DAGs).
a(n) = T(n, n-2), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=2.A026587
a(n) = T(n, n-2), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=2.
Duplicate of A026572.A026588
Duplicate of A026572.
a(n) = T(n,n-4), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.A026589
a(n) = T(n,n-4), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.
a(n) = T(2*n, n), where T is given by A026584.A026590
a(n) = T(2*n, n), where T is given by A026584.
a(n) = T(2*n, n-1), where T is given by A026584.A026591
a(n) = T(2*n, n-1), where T is given by A026584.
a(n) = T(2*n, n-2), where T is given by A026584.A026592
a(n) = T(2*n, n-2), where T is given by A026584.
a(n) = T(2*n-1, n-1), where T is given by A026584.A026593
a(n) = T(2*n-1, n-1), where T is given by A026584.
a(n) = T(2*n-1, n-2), where T is given by A026584.A026594
a(n) = T(2*n-1, n-2), where T is given by A026584.
a(n) = T(n, floor(n/2)), where T is given by A026584.A026595
a(n) = T(n, floor(n/2)), where T is given by A026584.
Row sums of A026584.A026596
Row sums of A026584.
Expansion of (1+x)/(1-x-4*x^2).A026597
Expansion of (1+x)/(1-x-4*x^2).
a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584.A026598
a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584.
a(n) = Sum_{j=0..2*i, i=0..n} A026584(i,j).A026599
a(n) = Sum_{j=0..2*i, i=0..n} A026584(i,j).
a(n) is the n-th letter of the infinite word generated from w(1)=1 inductively by w(n)=JUXTAPOSITION{w(n-1),w'(n-1),w"(n-1)}, where w(k) becomes w'(k) by the cyclic permutation 1->2->3->1 and w"(k) = (w')'(k).A026600
a(n) is the n-th letter of the infinite word generated from w(1)=1 inductively by w(n)=JUXTAPOSITION{w(n-1),w'(n-1),w"(n-1)}, where w(k) becomes w'(k) by the cyclic permutation 1->2->3->1 and w"(k) = (w')'(k).
Numbers k such that A026600(k) = 1.A026601
Numbers k such that A026600(k) = 1.
Numbers k such that A026600(k) = 2.A026602
Numbers k such that A026600(k) = 2.
Numbers k such that A026600(k) = 3.A026603
Numbers k such that A026600(k) = 3.
a(n) = s(1) + s(2) + ... + s(n), where s = A026600.A026604
a(n) = s(1) + s(2) + ... + s(n), where s = A026600.
[3->null]-transform of three-symbol Thue-Morse A026600.A026605
[3->null]-transform of three-symbol Thue-Morse A026600.
[1->null]-transform of three-symbol Thue-Morse A026600, with 1 subtracted.A026606
[1->null]-transform of three-symbol Thue-Morse A026600, with 1 subtracted.
Delete all 2's from A026600 and then replace each 3 with 2.A026607
Delete all 2's from A026600 and then replace each 3 with 2.
a(n) = number of 2's between n-th 1 and (n+1)st 1 in A026600.A026608
a(n) = number of 2's between n-th 1 and (n+1)st 1 in A026600.
a(n) = number of 3's between n-th 1 and (n+1)st 1 in A026600.A026609
a(n) = number of 3's between n-th 1 and (n+1)st 1 in A026600.
a(n) = number of 1's between n-th 2 and (n+1)st 2 in A026600.A026610
a(n) = number of 1's between n-th 2 and (n+1)st 2 in A026600.