Sequences

392,541 sequences

a(n) = Sum_{k=0..2n} (k+1) * A025177(n, k).A027261
a(n) = Sum_{k=0..2n} (k+1) * A025177(n, k).
a(n) = self-convolution of row n of array T given by A026519.A027262
a(n) = self-convolution of row n of array T given by A026519.
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026519.A027263
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026519.
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026519.A027264
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026519.
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026519.A027265
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026519.
a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).A027266
a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).
a(n) = self-convolution of row n of array T given by A026536.A027267
a(n) = self-convolution of row n of array T given by A026536.
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026536.A027268
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026536.
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026536.A027269
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026536.
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026536.A027270
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026536.
a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.A027271
a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.
Self-convolution of row n of array T given by A026552.A027272
Self-convolution of row n of array T given by A026552.
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026552.A027273
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026552.
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026552.A027274
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026552.
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026552.A027275
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026552.
a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).A027276
a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).
a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n-k,k).A027277
a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n-k,k).
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026568.A027278
a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026568.
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026568.A027279
a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026568.
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026568.A027280
a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026568.
a(n) = Sum_{k=0..2n} (k+1) * A026568(n, k).A027281
a(n) = Sum_{k=0..2n} (k+1) * A026568(n, k).
a(n) = self-convolution of row n of array T given by A026584.A027282
a(n) = self-convolution of row n of array T given by A026584.
a(n) = Sum_{k=0..2*n-1} T(n,k) * T(n,k+1), with T given by A026584.A027283
a(n) = Sum_{k=0..2*n-1} T(n,k) * T(n,k+1), with T given by A026584.
a(n) = Sum_{k=0..2*n-2} T(n,k) * T(n,k+2), with T given by A026584.A027284
a(n) = Sum_{k=0..2*n-2} T(n,k) * T(n,k+2), with T given by A026584.
a(n) = Sum_{k=0..2*n-3} T(n,k) * T(n,k+3), with T given by A026584.A027285
a(n) = Sum_{k=0..2*n-3} T(n,k) * T(n,k+3), with T given by A026584.
a(n) = Sum_{k=0..2n} (k+1) * A026584(n, k).A027286
a(n) = Sum_{k=0..2n} (k+1) * A026584(n, k).
Self-convolution of array T given by A026009.A027287
Self-convolution of array T given by A026009.
a(n) = Sum_{k=0..floor(n/2)} T(n,k) * T(n,k+1), with T given by A026009.A027288
a(n) = Sum_{k=0..floor(n/2)} T(n,k) * T(n,k+1), with T given by A026009.
a(n) = Sum_{k=0..floor(n/2)-1} T(n,k) * T(n,k+2), with T given by A026009.A027289
a(n) = Sum_{k=0..floor(n/2)-1} T(n,k) * T(n,k+2), with T given by A026009.
a(n) = Sum_{k=0..floor(n/2)-2} T(n,k) * T(n,k+3), with T given by A026009.A027290
a(n) = Sum_{k=0..floor(n/2)-2} T(n,k) * T(n,k+3), with T given by A026009.
a(n) = Sum_{k=0..floor(n/2)+1} (k+1) * A026009(n, k).A027291
a(n) = Sum_{k=0..floor(n/2)+1} (k+1) * A026009(n, k).
a(n) = Sum_{k=0..m} (k+1) * A026009(n, m-k) where m = floor(n/2)+1.A027292
a(n) = Sum_{k=0..m} (k+1) * A026009(n, m-k) where m = floor(n/2)+1.
Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.A027293
Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.
Self-convolution of row n of array T given by A026022.A027294
Self-convolution of row n of array T given by A026022.
a(n) = Sum_{k=0..m-1} T(n,k) * T(n,k+1), where m=n for n=0,1 and m=floor((n+3)/2) for n >= 2, and T given by A026022.A027295
a(n) = Sum_{k=0..m-1} T(n,k) * T(n,k+1), where m=n for n=0,1 and m=floor((n+3)/2) for n >= 2, and T given by A026022.
a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+2), with T given by A026022.A027296
a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+2), with T given by A026022.
a(n) = Sum_{k=0..floor((n-3)/2)} T(n,k) * T(n,k+3), with T given by A026022.A027297
a(n) = Sum_{k=0..floor((n-3)/2)} T(n,k) * T(n,k+3), with T given by A026022.
a(n) = Sum_{k=0..m} (k+1) * A026022(n, k), where m=n for n=0,1 and m = floor((n+3)/2) for n >= 2.A027298
a(n) = Sum_{k=0..m} (k+1) * A026022(n, k), where m=n for n=0,1 and m = floor((n+3)/2) for n >= 2.
a(n) = Sum_{k=0..m} (k+1) * A026022(n, m-k), where m=n for n=0,1 and m = floor((n+3)/2) for n >= 2.A027299
a(n) = Sum_{k=0..m} (k+1) * A026022(n, m-k), where m=n for n=0,1 and m = floor((n+3)/2) for n >= 2.
Triangular array Q given by rows: Q(n,k) = number of partitions of n that do not contain k as an element; domain: 1 <= k <= n, n >= 1.A027300
Triangular array Q given by rows: Q(n,k) = number of partitions of n that do not contain k as an element; domain: 1 <= k <= n, n >= 1.
a(n) = self-convolution of row n of Catalan triangle (A008315).A027301
a(n) = self-convolution of row n of Catalan triangle (A008315).
a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.A027302
a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.
a(n) = Sum_{k=0..floor((n-3)/2)} T(n,k) * T(n,k+2), with T given by A008315.A027303
a(n) = Sum_{k=0..floor((n-3)/2)} T(n,k) * T(n,k+2), with T given by A008315.
a(n) = Sum_{k=0..floor((n-5)/2)} T(n,k) * T(n,k+1), with T given by A008315.A027304
a(n) = Sum_{k=0..floor((n-5)/2)} T(n,k) * T(n,k+1), with T given by A008315.
a(n) = Sum_{k=0..floor((n+1)/2)} (k+1) * A008315(n, k).A027305
a(n) = Sum_{k=0..floor((n+1)/2)} (k+1) * A008315(n, k).
a(n) = 2^(n-1) + ((1 + (-1)^n)/4)*binomial(n, n/2).A027306
a(n) = 2^(n-1) + ((1 + (-1)^n)/4)*binomial(n, n/2).
Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).A027307
Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).
a(n) = self-convolution of row n of array T given by A026323.A027308
a(n) = self-convolution of row n of array T given by A026323.
a(n) = Sum_{k=0..n+1} T(n,k) * T(n,k+1), with T given by A026323.A027309
a(n) = Sum_{k=0..n+1} T(n,k) * T(n,k+1), with T given by A026323.
a(n) = Sum_{k=0..n} T(n,k) * T(n,k+2), with T given by A026323.A027310
a(n) = Sum_{k=0..n} T(n,k) * T(n,k+2), with T given by A026323.