-2024

Appears in sequences

• Table (read by rows) giving the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (k>=1, n>=2) contains T(i,k) for i=1 to k+1, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies Fact(n) = Sum_{i=1..k+1} T(i,k) * (n-1)^(k-i+1) / (2*k-2)!.at n=7A101751
• Odd triangle n!. This table read by rows gives the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.at n=8A102410
• a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.at n=24A105578
• Expansion of psi(-q^3) / f(q) where psi(), f() are Ramanujan theta functions.at n=23A139135
• The n-th term of the n-th Dirichlet self-convolution equals n^2.at n=45A163591
• G.f.: 1/(1-2*x+2*x^2-x^3+x^4).at n=27A199802
• Trisection 0 of A199802.at n=9A199927
• First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 219", based on the 5-celled von Neumann neighborhood.at n=25A270933
• Expansion of Product_{k>=1} (1 - q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).at n=35A304786
• First term of n-th difference sequence of (floor(Pi*k/3)), k >= 0.at n=24A325742
• Expansion of 1/(1 + x*Product_{k>=1} (1 - x^k)).at n=25A331484
• Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^4.at n=22A363598
• a(n) = n - A332215(n).at n=22A364253