A271662 Convolution of nonzero pentagonal numbers (A000326) with themselves.

1, 10, 49, 164, 434, 980, 1974, 3648, 6303, 10318, 16159, 24388, 35672, 50792, 70652, 96288, 128877, 169746, 220381, 282436, 357742, 448316, 556370, 684320, 834795, 1010646, 1214955, 1451044, 1722484, 2033104, 2387000, 2788544, 3242393, 3753498, 4327113, 4968804, 5684458

Offset

0,2

Comments

More generally, the ordinary generating function for the convolution of nonzero k-gonal numbers with themselves is (1 + (k - 3)*x)^2/(1 - x)^6.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Pentagonal Number

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

Formula

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

E.g.f.: (120 + 1080*x + 1800*x^2 + 920*x^3 + 165*x^4 + 9*x^5)*exp(x)/120.

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)*(9*n^2 + 21*n + 20)/120.

Sum_{n>=0} 1/a(n) = 1.13108002...

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 10, 49, 164, 434, 980}, 40]
Table[(n + 1) (n + 2) (n + 3) (9 n^2 + 21 n + 20)/120, {n, 0, 40}]
With[{nmax = 50}, CoefficientList[Series[(120 + 1080*x + 1800*x^2 + 920*x^3 + 165*x^4 + 9*x^5)*Exp[x]/120, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 07 2017 *)

Prog

(PARI) vector(40, n, n--; (n+1)*(n+2)*(n+3)*(9*n^2+21*n+20)/120) \\ Altug Alkan, Apr 12 2016
(Magma) /* From definition: */ P:=func<n, k | (n^2*(k-2)-n*(k-4))/2>; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 5)*P(i, 5): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 13 2016
(Magma) [(n+1)*(n+2)*(n+3)*(9*n^2+21*n+20)/120: n in [0..40]]; // Bruno Berselli, Apr 13 2016

Crossrefs

Cf. A000326.

Cf. similar sequences of the convolution of k-gonal numbers with themselves: A000389 (k=3, without zeros), A033455 (k=4), this sequence (k=5), A271870 (k=6).

Sequence in context: A307904 A226797 A163716 * A244864 A051947 A274561

Adjacent sequences: A271659 A271660 A271661 * A271663 A271664 A271665

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Apr 12 2016

Extensions

Edited by Bruno Berselli, Apr 13 2016

Status

approved