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 A300057 Coefficient of z^(3*n) in the expansion of (1 + 9*z + 9*z^2 + z^3)^(2*n). 2
 1, 164, 47148, 15454820, 5361965980, 1919987703504, 701459496193236, 259867456921970040, 97260263038893462300, 36686877800581349096240, 13924013746979490475444528, 5311128944356277793155688612, 2034235241375650519750351973188 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Muniru A Asiru, Table of n, a(n) for n = 0..100 FORMULA a(n) = 1/(2*Pi)*Integral_{0..2*Pi}(12*cos^2(x)*sin(x) + 20*sin^3(x))^(2*n) dx. a(n) = Sum_{k1=0..2*n} Sum_{k2=0..2*n} binomial(2*n,k1)*binomial(2*n,k2)*binomial(2*n,3*n-k1-k2)*((4-sqrt(15))^(2*n-k1))*((4+sqrt(15))^(2*n-k2)). a(n) = (c1/c3)*a(n-1)+(c2/c3)*a(n-2); with a(0)=1; a(1)=164; and c1=16*(n-1/2)*(-230+2259*n-3933*n^2+1863*n^3); c2=1036800*(n-1)*(n-3/2)*(n-1/2)*(n-1/9); c3=81*n*(n-2/3)*(n-1/3)*(n-10/9). From Wolfdieter Lang, Apr 06 2018: (Start) a(n) = 4^(2*n)*(2/Pi)*Integral_{0..Pi/2} sin(x)^(2*n)*(3 + 2*sin(x)^2)^(2*n) dx. With the binomial formula and integrals over even powers of sin(x) this becomes a(n) = 6^(2*n)*Sum_{k=0..2*n} binomial(2*n, k)*binomial(2*(n+k), n+k)*(1/6)^k = 6^(2*n)*binomial(2*n, n)*hypergeometric([-2*n, n+1/2], [n+1], -2/3). (End) a(n) ~ 2^(4*n) * 5^(2*n + 1/2) / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 18 2018 MATHEMATICA c1=16*(n-1/2)*(-230+2259*n-3933*n^2+1863*n^3); c2=1036800*(n-1)*(n-3/2)*(n-1/2)*(n-1/9); c3=81*n*(n-2/3)*(n-1/3)*(n-10/9); a[0]=1; a[1]=164; a[n0_]:=ReplaceAll[(c1/c3)*a[n0-1]+(c2/c3)*a[n0-2], n->n0] b[NN_]:=Expand[Total[Flatten[#]]&/@Table[Binomial[2*n, k2]*Binomial[2*n, k1]*Binomial[2*n, 3*n-k1-k2]*(4 + Sqrt[15])^(2*n-k1)*(4-Sqrt[15])^(2*n-k2), {n, 0, NN}, {k1, 0, 2*n}, {k2, 0, 2*n}]] ({#, SameQ[Coefficient[(1+9*z+9*z^2+z^3)^(2*#), z, 3*#]&/@Range[0, 10], #], SameQ[a/@Range[0, 10], #]}&@b[10])[[1]] Table[SeriesCoefficient[(1 + 9*z + 9*z^2 + z^3)^(2*n), {z, 0, 3*n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 18 2018 *) PROG (PARI) a(n) = polcoeff((1 + 9*z + 9*z^2 + z^3)^(2*n), 3*n); \\ Michel Marcus, Mar 06 2018 (GAP) List([0..15], n->6^(2*n)Sum([0..2*n], k->Binomial(2*n, k)*Binomial(2*(n+k), n+k)*(1/6)^k)); # Muniru A Asiru, Apr 07 2018 CROSSREFS Cf. A300058, A295870, A005721. Sequence in context: A035826 A238773 A289332 * A197440 A261758 A252725 Adjacent sequences: A300054 A300055 A300056 * A300058 A300059 A300060 KEYWORD nonn AUTHOR Bradley Klee, Feb 23 2018 STATUS approved

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Last modified October 1 22:38 EDT 2023. Contains 365828 sequences. (Running on oeis4.)