A277228 Convolution of the even-indexed triangular numbers (A014105) and the squares (A000290).

0, 0, 3, 22, 88, 258, 623, 1316, 2520, 4476, 7491, 11946, 18304, 27118, 39039, 54824, 75344, 101592, 134691, 175902, 226632, 288442, 363055, 452364, 558440, 683540, 830115, 1000818, 1198512, 1426278, 1687423, 1985488, 2324256, 2707760, 3140291, 3626406

Offset

0,3

Comments

This sequence was originally proposed in a comment on A071245 by J. M. Bergot as giving the first differences. Therefore, a(n) gives the partial sums of A071245.

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^2*(1 + x)*(3 + x)/(1 - x)^6 = (x*(3 + x)/(1 - x)^3)*(x*(1 + x)/(1 - x)^3).

a(n) = Sum_{k=0..n} A014105(n-k)*A000290(k).

a(n) = binomial(n+1, 3)*(4*n^2 + 5*n + 4)/10 = (n - 1)*n*(n + 1)*(4*n^2 + 5*n + 4)/60.

a(n) = Sum_{k=0..n} A071245(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 - 1) n (n + 1) (4 n^2 + 5 n + 4)/60, {n, 0, 40}] (* Bruno Berselli, Oct 21 2016 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 3, 22, 88, 258}, 40] (* Harvey P. Dale, Jun 04 2023 *)

Prog

(PARI) concat(vector(2), Vec(x^2*(1+x)*(3+x)/(1-x)^6 + O(x^50))) \\ Colin Barker, Oct 21 2016
(Magma) [Binomial(n+1, 3)*(4*n^2 +5*n +4)/10: n in [0..40]]; // G. C. Greubel, Oct 22 2018

Crossrefs

Cf. A000217, A000290, A014105, A071245.

Sequence in context: A004305 A275290 A368478 * A302272 A302723 A095663

Adjacent sequences: A277225 A277226 A277227 * A277229 A277230 A277231

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 20 2016

Status

approved