A216696 a(n) = Sum_{k=0..n} binomial(n,k)^4 * 2^k.

1, 3, 37, 495, 7761, 131283, 2336629, 43174911, 819869185, 15906350403, 313905320037, 6281740700271, 127173173346129, 2599950664710675, 53601450936173877, 1113117091905581055, 23262762639358582785, 488890438209132473475, 10325711607889973605285, 219057502101780979753455

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012

Formula

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.

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

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

Mathematica

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

Crossrefs

Cf. A005260.

Sequence in context: A336242 A054596 A155667 * A362724 A333385 A331114

Adjacent sequences: A216693 A216694 A216695 * A216697 A216698 A216699

Keyword

nonn

Author

Vaclav Kotesovec, Sep 15 2012

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

Status

approved