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

0,3

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).

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

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 *)

Prog

(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
(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
(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

Crossrefs

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

Sequence in context: A023421 A098051 A329053 * A100137 A325917 A210542

Adjacent sequences: A084634 A084635 A084636 * A084638 A084639 A084640

Keyword

easy,nonn

Author

Paul Barry, Jun 06 2003

Status

approved