A084636 Binomial transform of (1,0,1,0,1,0,2,0,2,0,2,0,...).

1, 1, 2, 4, 8, 16, 33, 71, 157, 349, 768, 1662, 3534, 7398, 15291, 31297, 63595, 128555, 258930, 520240, 1043540, 2090956, 4186757, 8379499, 16766313, 33541481, 67093588, 134199826, 268414602, 536846754, 1073713983, 2147451717, 4294930839, 8589893143

Offset: 0

Author

Paul Barry, Jun 06 2003

Keywords

• easy
• nonn

Comments

Partial sums are A084637 (without leading 1).

The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,2,2,2,...) with b(n) = Sum_{k=0..4} C(n,k) + 2*Sum_{k=5..n} C(n,k) = 2^(n+1) - (n^4 -2*n^3 + 11*n^2 + 14*n + 24)/24. This gives the partial sums of A084635.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-20,30,-25,11,-2).

Crossrefs

Cf. A000225, A000325, A084634, A084635, A084637.

Programs

Magma

[(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18) +0^n: n in [0..50]]; // G. C. Greubel, Mar 19 2023

Mathematica

Table[Boole[n==0] +(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18), {n,0,50}] (* G. C. Greubel, Mar 19 2023 *)

SageMath

[(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18) +0^n for n in range(51)] # G. C. Greubel, Mar 19 2023

Formula

a(n) = Sum_{k=0..2} C(n, 2*k) + 2*Sum_{k=3..floor(n/2)} C(n, 2*k).

a(n) = (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24 + 2*Sum_{k=3..floor(n/2)} C(n, 2*k).

O.g.f.: (1-2*x+2*x^2)*(1-4*x+5*x^2-2*x^3+x^4)/((1-x)^5*(1-2*x)). - R. J. Mathar, Apr 07 2008

a(n) = A000225(n) - (1/24)*n*(n-1)*(n^2 - 5*n + 18) + [n=0]. - G. C. Greubel, Mar 19 2023