A077044 Largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^5 = ((1-x^n)/(1-x))^5, i.e., the coefficient of x^floor(5*(n-1)/2) and of x^ceiling(5*(n-1)/2); also number of compositions of floor(5*(n+1)/2) into exactly 5 positive integers each no more than n.

0, 1, 10, 51, 155, 381, 780, 1451, 2460, 3951, 6000, 8801, 12435, 17151, 23030, 30381, 39280, 50101, 62910, 78151, 95875, 116601, 140360, 167751, 198780, 234131, 273780, 318501, 368235, 423851, 485250, 553401, 628160, 710601, 800530

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for sequences related to compositions

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Formula

a(n) = (230*n^4 + 70*n^2 + 27 - (30*n^2 + 27)*(-1)^n)/384 = A077042(n, 5).

a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8).

G.f.: -x*(1 + 8*x + 29*x^2 + 39*x^3 + 29*x^4 + 8*x^5 + x^6) / ( (1+x)^3*(x-1)^5 ). - R. J. Mathar, Sep 04 2011

Example

a(2)=10 since the compositions of floor(5*(2+1)/2) = 7 into exactly 5 positive integers each no more than 2 are: 1+1+1+2+2, 1+1+2+1+2, 1+1+2+2+1, 1+2+1+1+2, 1+2+1+2+1, 1+2+2+1+1, 2+1+1+1+2, 2+1+1+2+1, 2+1+2+1+1, 2+2+1+1+1.

Mathematica

LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 1, 10, 51, 155, 381, 780, 1451}, 40] (* Harvey P. Dale, Mar 05 2015 *)

Prog

(Magma) [(230*n^4+70*n^2+27-(30*n^2+27)*(-1)^n)/384: n in [0..40]]; // Vincenzo Librandi, Sep 05 2011
(PARI) a(n)=(230*n^4+70*n^2-30*n^2*(-1)^n)\/384 \\ Charles R Greathouse IV, Sep 25 2012

Crossrefs

Sequence in context: A264044 A143855 A124162 * A069038 A213563 A030183

Adjacent sequences: A077041 A077042 A077043 * A077045 A077046 A077047

Keyword

nonn,easy

Author

Henry Bottomley, Oct 22 2002

Status

approved