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

Author

Ilya Gutkovskiy, Apr 20 2016

Keywords

• nonn
• easy

Links

• 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).

Crossrefs

Cf. A000384.

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

Programs

Magma

[(n+1)*(n+2)*(n+3)*(4*n^2+6*n+5)/30 : n in [0..40]]; // Wesley Ivan Hurt, Apr 20 2016

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}]

PARI

a(n)=binomial(n+3,3)*(4*n^2 + 6*n + 5)/5 \\ Charles R Greathouse IV, Jul 26 2016

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.

Extensions

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