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A271663 Convolution of nonzero squares (A000290) with nonzero pentagonal numbers (A000326). 1
1, 9, 41, 131, 336, 742, 1470, 2682, 4587, 7447, 11583, 17381, 25298, 35868, 49708, 67524, 90117, 118389, 153349, 196119, 247940, 310178, 384330, 472030, 575055, 695331, 834939, 996121, 1181286, 1393016, 1634072, 1907400, 2216137, 2563617, 2953377, 3389163, 3874936 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More generally, the ordinary generating function for the convolution of nonzero h-gonal numbers and k-gonal numbers is (1 + (h - 3)*x)*(1 + (k - 3)*x)/(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, Square Number

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 + x)*(1 + 2*x)/(1 - x)^6.

E.g.f.: (120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*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)*(6*n^2 + 19*n + 20)/120.

Sum_{n>=0} 1/a(n) = 1.149165731...

MATHEMATICA

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 41, 131, 336, 742}, 40]

Table[(n + 1) (n + 2) (n + 3) (6 n^2 + 19 n + 20)/120, {n, 0, 40}]

With[{nmax = 50}, CoefficientList[Series[(120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*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)*(6*n^2+19*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, 4)*P(i, 5): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 12 2016

(MAGMA) [(n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120: n in [0..40]]; // Bruno Berselli, Apr 12 2016

CROSSREFS

Cf. A000290, A000326.

Cf. A005585: convolution of nonzero squares with nonzero triangular numbers.

Cf. A033455: convolution of nonzero squares with themselves.

Cf. A051836 (after 0): convolution of nonzero triangular numbers with nonzero pentagonal numbers.

Sequence in context: A297740 A297741 A001846 * A034441 A201275 A056243

Adjacent sequences:  A271660 A271661 A271662 * A271664 A271665 A271666

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Apr 12 2016

EXTENSIONS

Edited by Bruno Berselli, Apr 12 2016

STATUS

approved

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Last modified October 18 11:56 EDT 2018. Contains 316321 sequences. (Running on oeis4.)