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

0,3

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).

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

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 *)

Prog

(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
(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
(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

Crossrefs

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

Sequence in context: A117302 A265407 A023422 * A157021 A210543 A006211

Adjacent sequences: A084635 A084636 A084637 * A084639 A084640 A084641

Keyword

easy,nonn

Author

Paul Barry, Jun 06 2003

Status

approved