A216795 - OEIS

%I #18 Jul 11 2020 12:34:07

%S 1,4,58,1000,19426,412744,9195796,212836432,5062716850,123033947464,

%T 3041489363188,76243484446672,1933564156575364,49518970223489680,

%U 1278877982692134568,33269141292429734560,870987510534775369810,22930499187530338390600,606700679139764282611540

%N a(n) = sum_{k=0..n} binomial(n,k)^4 * 3^k.

%H Vincenzo Librandi, <a href="/A216795/b216795.txt">Table of n, a(n) for n = 0..200</a>

%H V. Kotesovec, <a href="http://www.kotesovec.cz/math_articles/kotesovec_binomial_asymptotics.pdf">Asymptotic of a sums of powers of binomial coefficients * x^k</a>, 2012

%F Recurrence: -(x-1)^4*(n+1)^3*(n+2)*(16*(4*x^2 + 17*x + 4)*n^4 + 184*(4*x^2 + 17*x + 4)*n^3 + (3137*x^2 + 13351*x + 3137)*n^2 + (5867*x^2 + 25051*x + 5867)*n + 4061*x^2 + 17438*x + 4061)*a(n) + (n+2)*(64*(4*x^5 + 141*x^4 + 655*x^3 + 655*x^2 + 141*x + 4)*n^7 + 1024*(4*x^5 + 141*x^4 + 655*x^3 + 655*x^2 + 141*x + 4)*n^6 + 4*(6857*x^5 + 242368*x^4 + 1126775*x^3 + 1126775*x^2 + 242368*x + 6857)*n^5 + 8*(12439*x^5 + 442336*x^4 + 2059985*x^3 + 2059985*x^2 + 442336*x + 12439)*n^4 + (211031*x^5 + 7579744*x^4 + 35400065*x^3 + 35400065*x^2 + 7579744*x + 211031)*n^3 + (261344*x^5 + 9524206*x^4 + 44667470*x^3 + 44667470*x^2 + 9524206*x + 261344)*n^2 + (174888*x^5 + 6498997*x^4 + 30655175*x^3 + 30655175*x^2 + 6498997*x + 174888)*n + 15*(3251*x^5 + 123835*x^4 + 588594*x^3 + 588594*x^2 + 123835*x + 3251))*a(n+1) -(32*(12*x^4-197*x^3-1030*x^2-197*x + 12)*n^8 + 624*(12*x^4-197*x^3-1030*x^2-197*x + 12)*n^7 + 10*(6295*x^4-103673*x^3-542004*x^2-103673*x + 6295)*n^6 + 4*(74418*x^4-1233343*x^3-6448430*x^2-1233343*x + 74418)*n^5 + (864893*x^4-14467663*x^3-75685660*x^2-14467663*x + 864893)*n^4 + 20*(78938*x^4-1336491*x^3-7002327*x^2-1336491*x + 78938)*n^3 + (1764932*x^4-30321697*x^3-159367410*x^2-30321697*x + 1764932)*n^2 + (1102551*x^4-19262286*x^3-101826010*x^2-19262286*x + 1102551)*n + 10*(29405*x^4-523232*x^3-2793306*x^2-523232*x + 29405))*a(n+2) + (n+3)*(64*(4*x^3 + 21*x^2 + 21*x + 4)*n^7 + 1152*(4*x^3 + 21*x^2 + 21*x + 4)*n^6 + 12*(2899*x^3 + 15226*x^2 + 15226*x + 2899)*n^5 + 4*(35609*x^3 + 187226*x^2 + 187226*x + 35609)*n^4 + (340693*x^3 + 1795162*x^2 + 1795162*x + 340693)*n^3 + (474743*x^3 + 2511132*x^2 + 2511132*x + 474743)*n^2 + (355831*x^3 + 1894439*x^2 + 1894439*x + 355831)*n + 10*(11039*x^3 + 59401*x^2 + 59401*x + 11039))*a(n+3) -(n+3)*(n+4)^3*(16*(4*x^2 + 17*x + 4)*n^4 + 120*(4*x^2 + 17*x + 4)*n^3 + (1313*x^2 + 5599*x + 1313)*n^2 + 15*(103*x^2 + 443*x + 103)*n + 659*x^2 + 2882*x + 659)*a(n+4) = 0, this is case x=3.

%F a(n) ~ (1+3^(1/4))^3/(4*sqrt(2)*3^(3/8)*Pi^(3/2)) * (1+3^(1/4))^(4*n)/n^(3/2). - _Vaclav Kotesovec_, Sep 19 2012

%F Generally, Sum_{k=0..n} binomial(n,k)^p*x^k is asymptotic a(n) ~ (1+x^(1/p))^(p*n+p-1)/sqrt((2*pi*n)^(p-1)*p*x^(1-1/p)). This is case p=4, x=3. - _Vaclav Kotesovec_, Sep 19 2012

%t Table[Sum[Binomial[n, k]^4*3^k, {k, 0, n}], {n, 0, 25}]

%Y Cf. A005260, A216696.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Sep 16 2012

%E Minor edits by _Vaclav Kotesovec_, Mar 31 2014