A277229 Convolution of the odd-indexed triangular numbers (A000384(n+1)) and the squares (A000290).

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

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