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A271870 Convolution of nonzero hexagonal numbers (A000384) with themselves.

%I A271870 #28 Feb 16 2025 08:33:33
%S A271870 1,12,66,236,651,1512,3108,5832,10197,16852,26598,40404,59423,85008,
%T A271870 118728,162384,218025,287964,374794,481404,610995,767096,953580,
%U A271870 1174680,1435005,1739556,2093742,2503396,2974791,3514656,4130192,4829088,5619537,6510252,7510482,8630028
%N A271870 Convolution of nonzero hexagonal numbers (A000384) with themselves.
%H A271870 OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H A271870 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>
%H A271870 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A271870 O.g.f.: (1 + 3*x)^2/(1 - x)^6.
%F A271870 E.g.f.: (30 + 330*x + 645*x^2 + 365*x^3 + 70*x^4 + 4*x^5)*exp(x)/30.
%F A271870 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 A271870 a(n) = (n + 1)*(n + 2)*(n + 3)*(4*n^2 + 6*n + 5)/30.
%p A271870 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 A271870 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 12, 66, 236, 651, 1512}, 36]
%t A271870 Table[(n + 1) (n + 2) (n + 3) ((4 n^2 + 6 n + 5)/30), {n, 0, 35}]
%o A271870 (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 A271870 (PARI) a(n)=binomial(n+3,3)*(4*n^2 + 6*n + 5)/5 \\ _Charles R Greathouse IV_, Jul 26 2016
%Y A271870 Cf. A000384.
%Y A271870 Cf. similar sequences of the convolution of k-gonal numbers with themselves listed in A271662.
%K A271870 nonn,easy
%O A271870 0,2
%A A271870 _Ilya Gutkovskiy_, Apr 20 2016
%E A271870 a(35)=8630028 corrected by _Georg Fischer_, Apr 03 2019