A271870 Convolution of nonzero hexagonal numbers (A000384) with themselves.

1, 12, 66, 236, 651, 1512, 3108, 5832, 10197, 16852, 26598, 40404, 59423, 85008, 118728, 162384, 218025, 287964, 374794, 481404, 610995, 767096, 953580, 1174680, 1435005, 1739556, 2093742, 2503396, 2974791, 3514656, 4130192, 4829088, 5619537, 6510252, 7510482, 8630028

Offset

0,2

Links

Table of n, a(n) for n=0..35.

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Hexagonal Number

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

Formula

O.g.f.: (1 + 3*x)^2/(1 - x)^6.

E.g.f.: (30 + 330*x + 645*x^2 + 365*x^3 + 70*x^4 + 4*x^5)*exp(x)/30.

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).

a(n) = (n + 1)*(n + 2)*(n + 3)*(4*n^2 + 6*n + 5)/30.

Maple

A271870:=n->(n+1)*(n+2)*(n+3)*(4*n^2+6*n+5)/30: seq(A271870(n), n=0..50); # Wesley Ivan Hurt, Apr 20 2016

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 12, 66, 236, 651, 1512}, 36]
Table[(n + 1) (n + 2) (n + 3) ((4 n^2 + 6 n + 5)/30), {n, 0, 35}]

Prog

(Magma) [(n+1)*(n+2)*(n+3)*(4*n^2+6*n+5)/30 : n in [0..40]]; // Wesley Ivan Hurt, Apr 20 2016
(PARI) a(n)=binomial(n+3, 3)*(4*n^2 + 6*n + 5)/5 \\ Charles R Greathouse IV, Jul 26 2016

Crossrefs

Cf. A000384.

Cf. similar sequences of the convolution of k-gonal numbers with themselves listed in A271662.

Sequence in context: A014787 A007249 A112142 * A114243 A000972 A180392

Adjacent sequences: A271867 A271868 A271869 * A271871 A271872 A271873

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Apr 20 2016

Extensions

a(35)=8630028 corrected by Georg Fischer, Apr 03 2019

Status

approved