A084637 Binomial transform of (1,0,1,0,1,0,1,1,1,1,1,...).

1, 1, 2, 4, 8, 16, 32, 65, 136, 293, 642, 1410, 3072, 6606, 14004, 29295, 60592, 124187, 252742, 511672, 1031912, 2075452, 4166408, 8353165, 16732664, 33498977, 67040458, 134134046, 268333872, 536748474, 1073595228, 2147309211, 4294760928, 8589691767

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, 2, 2, 2, ...) with A035038(n) = Sum_{k=0..5} C(n,k) + 2*Sum_{k=6..n} C(n,k) = 2^n - (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. This gives the partial sums of A084636.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-27,50,-55,36,-13,2).

Crossrefs

Cf. A000225, A000325, A035038, A084634, A084635, A084636.

Programs

Magma

[2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120: n in [0..50]]; // G. C. Greubel, Mar 19 2023

Mathematica

Table[2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120, {n,0,50}] (* G. C. Greubel, Mar 19 2023 *)

PARI

Vec((1-7*x+21*x^2-35*x^3+35*x^4-21*x^5+7*x^6)/((1-x)^6*(1-2*x)) + O(x^50)) \\ Colin Barker, Mar 17 2016

SageMath

[2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120 for n in range(51)] # G. C. Greubel, Mar 19 2023

Formula

a(n) = Sum_{k=0..2} C(n, 2*k) + Sum_{k=6..n} C(n, k).

a(n) = 2^n - n*(n^4 - 10*n^3 + 55*n^2 - 110*n + 184)/120.

G.f.: (1-7*x+21*x^2-35*x^3+35*x^4-21*x^5+7*x^6) / ((1-x)^6*(1-2*x)). - Colin Barker, Mar 17 2016