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
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Apr 12 2016
EXTENSIONS
Edited by Bruno Berselli, Apr 12 2016
STATUS
approved