%I #14 Sep 30 2022 01:45:29
%S 1,6,33,140,546,1932,6454,20448,62271,183202,523887,1461516,3991400,
%T 10698072,28203612,73265056,187822125,475788222,1192287117,2958453036,
%U 7274927646,17741533668,42937126290
%N Fifth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
%H G. C. Greubel, <a href="/A073375/b073375.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (6,-3,-40,45,126,-141,-252,180,320,-48,-192,-64).
%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073374(k).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+5, 5) * binomial(n-k, k) * 2^k.
%F a(n) = (n+3)*(n+9)*((3080 +1086*n +93*n^2)*(n+1)*U(n+1) + 2*(1660 +591*n +51*n^2) *(n+2)*U(n))/(5!*3^7) with U(n) = A001045(n+1), n>=0.
%F G.f.: 1/(1-(1+2*x)*x)^6 = 1/((1+x)*(1-2*x))^6.
%F E.g.f.: (1/787320)*(1024*(675 +3470*x +4195*x^2 +1830*x^3 +315*x^4 +18*x^5 )*exp(2*x) + (96120 -115640*x +42440*x^2 -6360*x^3 +405*x^4 -9*x^5)*exp(-x)). - _G. C. Greubel_, Sep 29 2022
%t Table[(n+3)*(n+9)*(2^(6+n)*(9*n^3 +117*n^2 +438*n +400) +(-1)^n*(9*n^3 +207*n^2 + 1518*n +3560))/787320, {n,0,40}] (* _G. C. Greubel_, Sep 29 2022 *)
%o (Magma) [(n+3)*(n+9)*(2^(6+n)*(9*n^3 +117*n^2 +438*n +400) +(-1)^n*(9*n^3 +207*n^2 + 1518*n +3560))/787320: n in [0..40]]; // _G. C. Greubel_, Sep 29 2022
%o (SageMath)
%o def A073375(n): return (n+3)*(n+9)*(2^(6+n)*(9*n^3 +117*n^2 +438*n +400) +(-1)^n*(9*n^3 +207*n^2 + 1518*n +3560))/787320
%o [A073375(n) for n in range(40)] # _G. C. Greubel_, Sep 29 2022
%Y Sixth (m=5) column of triangle A073370.
%Y Cf. A001045, A073374.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 02 2002