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%I #27 Sep 08 2022 08:46:16
%S 1,12,66,236,651,1512,3108,5832,10197,16852,26598,40404,59423,85008,
%T 118728,162384,218025,287964,374794,481404,610995,767096,953580,
%U 1174680,1435005,1739556,2093742,2503396,2974791,3514656,4130192,4829088,5619537,6510252,7510482,8630028
%N Convolution of nonzero hexagonal numbers (A000384) with themselves.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F O.g.f.: (1 + 3*x)^2/(1 - x)^6.
%F E.g.f.: (30 + 330*x + 645*x^2 + 365*x^3 + 70*x^4 + 4*x^5)*exp(x)/30.
%F 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 a(n) = (n + 1)*(n + 2)*(n + 3)*(4*n^2 + 6*n + 5)/30.
%p A271870:=n->(n+1)*(n+2)*(n+3)*(4*n^2+6*n+5)/30: seq(A271870(n), n=0..50); # _Wesley Ivan Hurt_, Apr 20 2016
%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 12, 66, 236, 651, 1512}, 36]
%t Table[(n + 1) (n + 2) (n + 3) ((4 n^2 + 6 n + 5)/30), {n, 0, 35}]
%o (Magma) [(n+1)*(n+2)*(n+3)*(4*n^2+6*n+5)/30 : n in [0..40]]; // _Wesley Ivan Hurt_, Apr 20 2016
%o (PARI) a(n)=binomial(n+3,3)*(4*n^2 + 6*n + 5)/5 \\ _Charles R Greathouse IV_, Jul 26 2016
%Y Cf. A000384.
%Y Cf. similar sequences of the convolution of k-gonal numbers with themselves listed in A271662.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Apr 20 2016
%E a(35)=8630028 corrected by _Georg Fischer_, Apr 03 2019
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