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

Author

Ilya Gutkovskiy, Apr 12 2016

Keywords

• nonn
• easy

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)

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).

Programs

Magma

/* From definition: */ P:=func; /*, 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

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 *)

PARI

vector(40, n, n--; (n+1)*(n+2)*(n+3)*(9*n^2+21*n+20)/120) \\ Altug Alkan, Apr 12 2016

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...

Extensions

Edited by Bruno Berselli, Apr 13 2016