A271663 Convolution of nonzero squares (A000290) with nonzero pentagonal numbers (A000326).

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

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2016

Keywords

• nonn
• easy

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)

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.

Programs

Magma

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

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 *)

PARI

vector(40, n, n--; (n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120) \\ Altug Alkan, Apr 12 2016

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...

Extensions

Edited by Bruno Berselli, Apr 12 2016