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

0,3

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).

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A299271 A110334 A357904 * A352044 A161869 A210541

Adjacent sequences: A084633 A084634 A084635 * A084637 A084638 A084639

Keyword

easy,nonn

Author

Paul Barry, Jun 06 2003

Status

approved