A181069 Expansion of l.g.f. Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *x^k ] *x^n/n.

1, 3, 28, 275, 3126, 37632, 475056, 6192531, 82754650, 1127504378, 15603575208, 218727171104, 3099183987004, 44315462038200, 638663235342528, 9267264584278419, 135279095477748642, 1985221072388231742

Offset: 1

Author

Paul D. Hanna, Oct 08 2010

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 +...
which equals the series:
  L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
  + (1 + 2^4*x +  3^4*x^2 +  4^4*x^3 +   5^4*x^4 +   6^4*x^5 + ...)*x^2/2
  + (1 + 3^4*x +  6^4*x^2 + 10^4*x^3 +  15^4*x^4 +  21^4*x^5 + ...)*x^3/3
  + (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 +  35^4*x^4 +  56^4*x^5 + ...)*x^4/4
  + (1 + 5^4*x + 15^4*x^2 + 35^4*x^3 +  70^4*x^4 + 126^4*x^5 + ...)*x^5/5
  + (1 + 6^4*x + 21^4*x^2 + 56^4*x^3 + 126^4*x^4 + 252^4*x^5 + ...)*x^6/6
  + (1 + 7^4*x + 28^4*x^2 + 84^4*x^3 + 210^4*x^4 + 462^4*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181068:
  exp(L(x)) = 1 + x + 2*x^2 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A181067 (variant), A181068.

Cf. A005260, A198256.

Programs

Magma

[(&+[Binomial(n,k)*Binomial(n-1, k)^3: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Apr 05 2021

Maple

A181069:= n-> add( binomial(n,k)*binomial(n-1,k)^3, k=0..n-1); seq(A181069(n), n=1..20); # G. C. Greubel, Apr 05 2021

Mathematica

Table[Sum[Binomial[n-1,k]^3 * Binomial[n,k],{k,0,n-1}],{n,1,20}] (* Vaclav Kotesovec, Mar 06 2014 *)
a[n_] := HypergeometricPFQ[{-n + 1, -n + 1, -n + 1, -n}, {1, 1, 1}, 1];Table[a[n], {n, 1, 18}] (* Detlef Meya, May 28 2024 *)

PARI

{a(n)=sum(k=0, n-1, binomial(n-1, k)^4*n/(n-k))}

PARI

{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^4*x^k)*x^m/m)+x*O(x^n), n)}
for(n=1,20,print1(a(n),", "))

Sage

[sum( binomial(n,k)*binomial(n-1,k)^3 for k in (0..n-1) ) for n in (1..20)] # G. C. Greubel, Apr 05 2021

Formula

a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * binomial(n,k).

From Vaclav Kotesovec, Mar 06 2014: (Start)

Recurrence: (n-1)^2*n^3*(10*n^2 - 25*n + 16)*a(n) = 2*(n-1)^2*(60*n^5 - 240*n^4 + 341*n^3 - 225*n^2 + 90*n - 16)*a(n-1) + 4*(n-2)^2*n*(4*n - 7)*(4*n - 5)*(10*n^2 - 5*n + 1)*a(n-2).

a(n) ~ 2^(4*n-5/2) / (Pi*n)^(3/2). (End)

a(n) = hypergeom([-n + 1, -n + 1, -n + 1, -n], [1, 1, 1], 1). - Detlef Meya, May 28 2024

a(n) = Sum_{k=0..n} (k/n)^3 * binomial(n,k)^4. - Seiichi Manyama, Jul 14 2024