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A277229 Convolution of the odd-indexed triangular numbers (A000384(n+1)) and the squares (A000290). 2
0, 1, 10, 48, 158, 413, 924, 1848, 3396, 5841, 9526, 14872, 22386, 32669, 46424, 64464, 87720, 117249, 154242, 200032, 256102, 324093, 405812, 503240, 618540, 754065, 912366, 1096200, 1308538, 1552573, 1831728, 2149664, 2510288, 2917761, 3376506, 3891216 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This sequence was originally proposed in a comment on A071238 by J. M. Bergot as giving the first differences. Therefore, a(n) gives the partial sums of A071238.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

FORMULA

O.g.f.: x*(1 + x)*(1 + 3*x)/(1 - x)^6 = ((1 + 3*x)/(1 - x)^3)*(x*(1 + x)/(1 - x)^3).

a(n) = Sum_{k=0..n} A000384(n+1-k)*A000290(k).

a(n) = binomial(n+2, 3)*(4*n^2 + 3*n + 3)/10 = n*(n + 1)*(n + 2)*(4*n^2 + 3*n + 3)/60.

a(n) = Sum_{k=0..n} A071238(k).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. - Colin Barker, Oct 21 2016

MATHEMATICA

Table[n (n + 1) (n + 2) (4 n^2 + 3 n + 3)/60, {n, 0, 40}] (* Bruno Berselli, Oct 21 2016 *)

PROG

(PARI) concat(0, Vec(x*((1+x)*(1+3*x))/(1-x)^6 + O(x^50))) \\ Colin Barker, Oct 21 2016

CROSSREFS

Cf. A000217, A000290, A000384, A071238.

Sequence in context: A210371 A195023 A353620 * A163724 A271638 A238916

Adjacent sequences:  A277226 A277227 A277228 * A277230 A277231 A277232

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Oct 20 2016

STATUS

approved

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Last modified August 13 18:03 EDT 2022. Contains 356107 sequences. (Running on oeis4.)