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%I #22 May 15 2026 16:48:15
%S 0,0,0,1,12,99,688,4326,25464,143018,775536,4092207,21126564,
%T 107136597,535272864,2640788828,12888228336,62312915172,298810304672,
%U 1422537514989,6728698090428,31643981809111,148044975160272,689370668876130,3196344091219560,14762394097629870
%N Triple convolution of the modified Fibonacci sequence F(3n)/2 (A001076).
%H D. Dmytryshyn, D. Gray, V. Khamitov, and Alex Stokolos, <a href="https://arxiv.org/abs/2603.08636">Convolved numbers of k-section of the Fibonacci sequence: properties, consequences</a>, arXiv:2603.08636 [math.CA], 2026; see also <a href="https://doi.org/10.15276/hait.09.2026.09">alternate link</a>, Herald of Advanced Information Technology. 2026. Vol. 9, No. 2. P. 129-139.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,-45,40,45,12,1).
%F a(n) = Sum_{j=1..n} y(j)*y1(n-j), where y(n) = F(3*n)/2 and y1(n) is the first-order convolution of y.
%F G.f.: x^3/(1 - 4*x - x^2)^3.
%F a(n) = (-1)^(n-3) * GegenbauerC(n-3, 3, 2*I).
%F a(j) = (4*(j-1)*a(j-1) + (j+1)*a(j-2)) / (j-3) for j>=4, with a(0..2)=0, a(3)=1.
%F a(n) = 12*a(n-1) - 45*a(n-2) + 40*a(n-3) + 45*a(n-4) + 12*a(n-5) + a(n-6) for n>=6, with a(0..5) = (0, 0, 0, 1, 12, 99).
%F a(n) = (1/800) * ( 1/2*(n-2)*(n-1)*F(3*n+6) + (n-2)*(n+2)*F(3*n) + 1/2*(n+1)*(n+2)*F(3*n-6) ).
%F a(n) = (q^(-3*n-6) / (q^3+q^(-3))^5) * ( 1/2*(n-2)*(n-1)*(q^(6*n+12) - (-1)^n) + (n-2)*(n+2)*(q^(6*n+6) - (-1)^n*q^6) + 1/2*(n+1)*(n+2)*(q^(6*n) - (-1)^n*q^12) ), where q = (1+sqrt(5))/2.
%F E.g.f.: exp(2*x) * ( (1/10 - 7/200*x)*cosh(sqrt(5)*x) + sqrt(5)*(9/200*x^2 - 1/50*x + 7/1000)*sinh(sqrt(5)*x) ).
%p N := 25; a[0]:= 0; a[1]:= 0; a[2]:= 0; a[3]:= 1;
%p for j from 4 to N do
%p a[j]:=1/(j-3)*(4*(j-1)*a[j-1]+(j+1)*a[j-2]):
%p end:
%p seq(a[k], k = 0 .. N);
%t LinearRecurrence[{12, -45, 40, 45, 12, 1}, {0, 0, 0, 1, 12, 99}, 26] (* _Amiram Eldar_, May 10 2026 *)
%Y Cf. A395431, A001076, A000045, A001622.
%K nonn,easy
%O 0,5
%A _Alex Stokolos_, May 08 2026