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%I #21 Sep 30 2022 02:04:42
%S 1,3,12,34,99,261,678,1692,4149,9959,23568,55014,127031,290457,658602,
%T 1482240,3314025,7365915,16285300,35832810,78500811,171293293,
%U 372412782,806963364,1743173469,3754782351,8066319768,17285917742,36957928479,78847115649
%N Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
%H G. C. Greubel, <a href="/A073372/b073372.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-11,-6,12,8).
%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073371(k).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2) * binomial(n-k, k) * 2^k.
%F a(n) = ((30+9*n)*(n+1)*U(n+1) + 2*(33+9*n)*(n+2)*U(n))/162 with U(n) = A001045(n+1), n>=0.
%F G.f.: 1/(1 - (1+2*x)*x)^3.
%F E.g.f.: (1/162)*(32*(4 + 9*x + 3*x^2)*exp(2*x) + (34 - 24*x + 3*x^2)*exp(-x)). - _G. C. Greubel_, Sep 28 2022
%t CoefficientList[Series[-(-1+x+2x^2)^(-3),{x,0,78}],x] (* or *) Table[(-3*(-1)^n*n^2+3*2^(n+2)*n^2-15*(-1)^n*n+9*2^(n+2)*n-16*(-1)^n+2^(n+4))/162,{n,42}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2012 *)
%o (Magma) [(2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162: n in [0..40]]; // _G. C. Greubel_, Sep 28 2022
%o (SageMath)
%o def A073372(n): return (2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162
%o [A073372(n) for n in range(40)] # _G. C. Greubel_, Sep 28 2022
%Y Third (m=2) column of triangle A073370.
%Y Cf. A001045, A073371.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 02 2002
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