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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 24 18:12 EDT 2019. Contains 326295 sequences. (Running on oeis4.)