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A271662 Convolution of nonzero pentagonal numbers (A000326) with themselves. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A056578 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

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Last modified October 17 07:05 EDT 2018. Contains 316276 sequences. (Running on oeis4.)