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A279159
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a(n) = Sum_{k=0..n} (k+1)*binomial(3*n-4*k+1,n-k)/(n-k+1).
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1
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1, 4, 13, 47, 202, 982, 5119, 27819, 155545, 888517, 5162089, 30406466, 181164254, 1089858766, 6610791274, 40386823863, 248278003097, 1534716252481, 9533278498468, 59477997359141, 372545818015840, 2341796524294495, 14768075460084253
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: -1/(1 - x*u(x)) + 1/(1 - x)/x^2, where u(x) = 1-(4/3)*sin(asin(3^(3/2)*sqrt(x)/2)/3)^2.
Recurrence: 2*(n + 1)*(2*n + 1)*(1955*n^5 - 17254*n^4 + 58537*n^3 - 94766*n^2 + 72288*n - 20160)*a(n) = 5*(13685*n^7 - 116086*n^6 + 370418*n^5 - 538048*n^4 + 319601*n^3 - 12874*n^2 - 36120*n + 4032)*a(n-1) - 2*(56695*n^7 - 494501*n^6 + 1650289*n^5 - 2596715*n^4 + 1841560*n^3 - 309704*n^2 - 216024*n + 80640)*a(n-2) + 3*(3*n - 8)*(3*n - 7)*(1955*n^5 - 7479*n^4 + 9071*n^3 - 3129*n^2 - 874*n + 600)*a(n-3) - 2*(n + 1)*(2*n + 1)*(1955*n^5 - 17254*n^4 + 58537*n^3 - 94766*n^2 + 72288*n - 20160)*a(n-4) + (60605*n^7 - 523144*n^6 + 1717556*n^5 - 2627890*n^4 + 1760375*n^3 - 227926*n^2 - 204216*n + 60480)*a(n-5) - 3*(3*n - 8)*(3*n - 7)*(1955*n^5 - 7479*n^4 + 9071*n^3 - 3129*n^2 - 874*n + 600)*a(n-6).
a(n) ~ 3^(3*n + 19/2) / (5329 * sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). (End)
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MATHEMATICA
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Table[Sum[(k+1)*Binomial[3*n-4*k+1, n-k]/(n-k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 07 2016 *)
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PROG
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(Maxima) u(x):=1-(4/3)*sin((1/3)*asin(sqrt(27*x/4)))^2; taylor(-1/(1 - x*u(x)) + 1/(1 - x)/x^2, x, 0, 27);
(PARI) for(n=0, 25, print1(sum(k=0, n, (k+1)*binomial(3*n - 4*k +1, n-k)/(n-k+1)), ", ")) \\ G. C. Greubel, Jun 06 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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