A206180 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 3^k.

1, 4, 34, 352, 3946, 46744, 573616, 7217536, 92527738, 1203467464, 15834369244, 210304283776, 2815055712496, 37930536447808, 513972867056704, 6998587355233792, 95704396144575898, 1313665229153722408, 18091969874675059204, 249908773119244105792

Offset: 0

Author

Paul D. Hanna, Feb 04 2012

Keywords

• nonn

Comments

Ignoring initial term, equals the logarithmic derivative of A206179.

Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.

Diagonal of rational function 1/(1 + y + z + x*y + y*z + 3*x*z + 4*x*y*z). - Gheorghe Coserea, Jul 01 2018

Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 3*x*y*z). - Seiichi Manyama, Jul 11 2020

Examples

			L.g.f.: L(x) = 4*x + 34*x^2/2 + 352*x^3/3 + 3946*x^4/4 + 46744*x^5/5 +...
Exponentiation equals the g.f. of A206179:
exp(L(x)) = 1 + 4*x + 25*x^2 + 196*x^3 + 1747*x^4 + 16996*x^5 + 175936*x^6 +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Crossrefs

Cf. A206179, A000172, A206178, A216483, A216636.

Programs

Mathematica

Flatten[{1,RecurrenceTable[{(n+3)^2*(3*n+4)*a[n+3]-4*(9*n^3+57*n^2+116*n+74)*a[n+2]-(99*n^3+528*n^2+929*n+540)*a[n+1]-64*(3*n+7)*(n+1)^2*a[n]==0,a[1]==4,a[2]==34,a[3]==352},a,{n,1,20}]}] (* Vaclav Kotesovec, Sep 11 2012 *)
Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -3] , {n, 0, 20}] (* Jean-François Alcover, Oct 25 2019 *)

PARI

{a(n)=sum(k=0,n,binomial(n,k)^3*3^k)}
for(n=0,41,print1(a(n),", "))

Sage

A206180 = lambda n: hypergeometric([-n,-n,-n],[1,1], -3)
[Integer(A206180(n).n(100)) for n in (0..19)] # Peter Luschny, Sep 23 2014

Formula

Recurrence: (n+3)^2*(3*n+4)*a(n+3) - 4*(9*n^3+57*n^2+116*n+74)*a(n+2) - (99*n^3+528*n^2+929*n+540)*a(n+1) - 64*(3*n+7)*(n+1)^2*a(n) = 0. - Vaclav Kotesovec, Sep 11 2012

a(n) ~ (1 + 3^(1/3))^(3*n + 2) / (2*3^(5/6)*Pi*n). - Vaclav Kotesovec, Sep 19 2012, simplified Apr 24 2025

G.f.: hypergeom([1/3, 2/3],[1],81*x^2/(1-4*x)^3)/(1-4*x). - Mark van Hoeij, May 02 2013

a(n) = hypergeometric([-n,-n,-n],[1,1], -3). - Peter Luschny, Sep 23 2014

G.f. y=A(x) satisfies: 0 = x*(2*x + 1)*(64*x^3 + 33*x^2 + 12*x - 1)*y'' + (384*x^4 + 388*x^3 + 123*x^2 + 24*x - 1)*y' + (128*x^3 + 132*x^2 + 24*x + 4)*y. - Gheorghe Coserea, Jul 01 2018

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014