A084638 Binomial transform of (1,0,1,0,1,0,1,0,2,0,2,0,2,....).

1, 1, 2, 4, 8, 16, 32, 64, 129, 265, 558, 1200, 2610, 5682, 12288, 26292, 55587, 116179, 240366, 493108, 1004780, 2036692, 4112144, 8278552, 16631717, 33364381, 66863358, 133903816, 268037862, 536371734, 1073120208, 2146715436, 4294024647, 8588785575

Offset: 0

Author

Paul Barry, Jun 06 2003

Keywords

• easy
• nonn

Comments

The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,1,1,2,2...) with a(n) = Sum_{k=0..6} C(n,k) + 2*Sum_{k=7..n} C(n,k) = 2^(n+1) - A008859(n). This gives the partial sums of A084637.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-35,77,-105,91,-49,15,-2).

Crossrefs

Cf. A000225, A000325, A008859, A084634, A084635, A084636, A084637.

Programs

Magma

[2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n: n in [0..50]]; // G. C. Greubel, Mar 20 2023

Mathematica

Table[2^n -4 -(1/6!)*(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160) + Boole[n==0], {n,0,50}] (* G. C. Greubel, Mar 20 2023 *)

PARI

Vec((1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8)/((1-x)^7*(1-2*x)) + O(x^50)) \\ Colin Barker, Mar 17 2016

SageMath

[2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n for n in range(51)] # G. C. Greubel, Mar 20 2023

Formula

a(n) = Sum_{k=0..3, C(n, 2*k)} + 2*Sum_{k=4..floor(n/2), C(n, 2*k)}.

a(n) = (n^6-15*n^5+115*n^4-405*n^3+964*n^2-660*n+720)/720 + 2*Sum_{k=4..floor(n/2), C(n, 2k)}.

G.f.: (1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8) / ((1-x)^7*(1-2*x)). - Colin Barker, Mar 17 2016