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A271662 Convolution of nonzero pentagonal numbers (A000326) with themselves.

%I A271662 #33 Feb 16 2025 08:33:33
%S A271662 1,10,49,164,434,980,1974,3648,6303,10318,16159,24388,35672,50792,
%T A271662 70652,96288,128877,169746,220381,282436,357742,448316,556370,684320,
%U A271662 834795,1010646,1214955,1451044,1722484,2033104,2387000,2788544,3242393,3753498,4327113,4968804,5684458
%N A271662 Convolution of nonzero pentagonal numbers (A000326) with themselves.
%C A271662 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.
%H A271662 G. C. Greubel, <a href="/A271662/b271662.txt">Table of n, a(n) for n = 0..1000</a>
%H A271662 OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H A271662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>
%H A271662 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1)
%F A271662 O.g.f.: (1 + 2*x)^2/(1 - x)^6.
%F A271662 E.g.f.: (120 + 1080*x + 1800*x^2 + 920*x^3 + 165*x^4 + 9*x^5)*exp(x)/120.
%F A271662 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 A271662 a(n) = (n + 1)*(n + 2)*(n + 3)*(9*n^2 + 21*n + 20)/120.
%F A271662 Sum_{n>=0} 1/a(n) = 1.13108002...
%t A271662 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 10, 49, 164, 434, 980}, 40]
%t A271662 Table[(n + 1) (n + 2) (n + 3) (9 n^2 + 21 n + 20)/120, {n, 0, 40}]
%t A271662 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 *)
%o A271662 (PARI) vector(40, n, n--; (n+1)*(n+2)*(n+3)*(9*n^2+21*n+20)/120) \\ _Altug Alkan_, Apr 12 2016
%o A271662 (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
%o A271662 (Magma) [(n+1)*(n+2)*(n+3)*(9*n^2+21*n+20)/120: n in [0..40]]; // _Bruno Berselli_, Apr 13 2016
%Y A271662 Cf. A000326.
%Y A271662 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).
%K A271662 nonn,easy
%O A271662 0,2
%A A271662 _Ilya Gutkovskiy_, Apr 12 2016
%E A271662 Edited by _Bruno Berselli_, Apr 13 2016