A279159 - OEIS

%I #18 Jun 07 2017 00:43:09

%S 1,4,13,47,202,982,5119,27819,155545,888517,5162089,30406466,

%T 181164254,1089858766,6610791274,40386823863,248278003097,

%U 1534716252481,9533278498468,59477997359141,372545818015840,2341796524294495,14768075460084253

%N a(n) = Sum_{k=0..n} (k+1)*binomial(3*n-4*k+1,n-k)/(n-k+1).

%H G. C. Greubel, <a href="/A279159/b279159.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

%F From _Vaclav Kotesovec_, Dec 07 2016: (Start)

%F 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).

%F a(n) ~ 3^(3*n + 19/2) / (5329 * sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). (End)

%t 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 *)

%o (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);

%o (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

%Y Cf. A006013.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Dec 07 2016