A216696 - OEIS

%I #26 Jul 11 2020 12:37:43

%S 1,3,37,495,7761,131283,2336629,43174911,819869185,15906350403,

%T 313905320037,6281740700271,127173173346129,2599950664710675,

%U 53601450936173877,1113117091905581055,23262762639358582785,488890438209132473475,10325711607889973605285,219057502101780979753455

%N a(n) = Sum_{k=0..n} binomial(n,k)^4 * 2^k.

%H Vincenzo Librandi, <a href="/A216696/b216696.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: (864*n^8 + 14256*n^7 + 99843*n^6 + 386844*n^5 + 905129*n^4 + 1307419*n^3 + 1137462*n^2 + 545141*n + 110362)*a(n) - (673920*n^8 + 12130560*n^7 + 94006260*n^6 + 409620480*n^5 + 1097677875*n^4 + 1852470090*n^3 + 1922754750*n^2 + 1122222315*n + 281983230)*a(n+1) - (188352*n^8 + 3672864*n^7 + 30977310*n^6 + 147448176*n^5 + 432716089*n^4 + 800645440*n^3 + 910682766*n^2 + 581183533*n + 159056590)*a(n+2) - (10368*n^8 + 217728*n^7 + 1969236*n^6 + 10003440*n^5 + 31163253*n^4 + 60851106*n^3 + 72587550*n^2 + 48264909*n + 13672710)*a(n+3) + (864*n^8 + 19440*n^7 + 187539*n^6 + 1011492*n^5 + 3330104*n^4 + 6840009*n^3 + 8542572*n^2 + 5919152*n + 1739328)*a(n+4) = 0.

%F a(n) ~ (1+2^(1/4))^3/(4*2^(7/8)*Pi^(3/2)) * (1+2^(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=2. - _Vaclav Kotesovec_, Sep 19 2012

%t Table[Sum[Binomial[n, k]^4*2^k, {k, 0, n}], {n, 0, 25}]

%Y Cf. A005260.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Sep 15 2012

%E Minor edits by _Vaclav Kotesovec_, Mar 31 2014