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

%I A271663 #40 Feb 16 2025 08:33:33
%S A271663 1,9,41,131,336,742,1470,2682,4587,7447,11583,17381,25298,35868,49708,
%T A271663 67524,90117,118389,153349,196119,247940,310178,384330,472030,575055,
%U A271663 695331,834939,996121,1181286,1393016,1634072,1907400,2216137,2563617,2953377,3389163,3874936
%N A271663 Convolution of nonzero squares (A000290) with nonzero pentagonal numbers (A000326).
%C A271663 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.
%H A271663 G. C. Greubel, <a href="/A271663/b271663.txt">Table of n, a(n) for n = 0..1000</a>
%H A271663 OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H A271663 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A271663 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>
%H A271663 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1)
%F A271663 O.g.f.: (1 + x)*(1 + 2*x)/(1 - x)^6.
%F A271663 E.g.f.: (120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*x^5)*exp(x)/120.
%F A271663 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).
%F A271663 a(n) = (n + 1)*(n + 2)*(n + 3)*(6*n^2 + 19*n + 20)/120.
%F A271663 Sum_{n>=0} 1/a(n) = 1.149165731...
%t A271663 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 41, 131, 336, 742}, 40]
%t A271663 Table[(n + 1) (n + 2) (n + 3) (6 n^2 + 19 n + 20)/120, {n, 0, 40}]
%t A271663 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 *)
%o A271663 (PARI) vector(40, n, n--; (n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120) \\ _Altug Alkan_, Apr 12 2016
%o A271663 (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
%o A271663 (Magma) [(n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120: n in [0..40]]; // _Bruno Berselli_, Apr 12 2016
%Y A271663 Cf. A000290, A000326.
%Y A271663 Cf. A005585: convolution of nonzero squares with nonzero triangular numbers.
%Y A271663 Cf. A033455: convolution of nonzero squares with themselves.
%Y A271663 Cf. A051836 (after 0): convolution of nonzero triangular numbers with nonzero pentagonal numbers.
%K A271663 nonn,easy
%O A271663 0,2
%A A271663 _Ilya Gutkovskiy_, Apr 12 2016
%E A271663 Edited by _Bruno Berselli_, Apr 12 2016