A181069 - OEIS

%I #25 Sep 22 2025 16:01:04

%S 1,3,28,275,3126,37632,475056,6192531,82754650,1127504378,15603575208,

%T 218727171104,3099183987004,44315462038200,638663235342528,

%U 9267264584278419,135279095477748642,1985221072388231742

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

%H G. C. Greubel, <a href="/A181069/b181069.txt">Table of n, a(n) for n = 1..500</a>

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

%F From _Vaclav Kotesovec_, Mar 06 2014: (Start)

%F 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).

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

%F a(n) = hypergeom([-n + 1, -n + 1, -n + 1, -n], [1, 1, 1], 1). - _Detlef Meya_, May 28 2024

%F a(n) = Sum_{k=0..n} (k/n)^3 * binomial(n,k)^4. - _Seiichi Manyama_, Jul 14 2024

%e L.g.f.: L(x) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 +...

%e which equals the series:

%e   L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x

%e   + (1 + 2^4*x +  3^4*x^2 +  4^4*x^3 +   5^4*x^4 +   6^4*x^5 + ...)*x^2/2

%e   + (1 + 3^4*x +  6^4*x^2 + 10^4*x^3 +  15^4*x^4 +  21^4*x^5 + ...)*x^3/3

%e   + (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 +  35^4*x^4 +  56^4*x^5 + ...)*x^4/4

%e   + (1 + 5^4*x + 15^4*x^2 + 35^4*x^3 +  70^4*x^4 + 126^4*x^5 + ...)*x^5/5

%e   + (1 + 6^4*x + 21^4*x^2 + 56^4*x^3 + 126^4*x^4 + 252^4*x^5 + ...)*x^6/6

%e   + (1 + 7^4*x + 28^4*x^2 + 84^4*x^3 + 210^4*x^4 + 462^4*x^5 + ...)*x^7/7 + ...

%e Exponentiation yields the g.f. of A181068:

%e   exp(L(x)) = 1 + x + 2*x^2 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +...

%p 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

%t Table[Sum[Binomial[n-1,k]^3 * Binomial[n,k],{k,0,n-1}],{n,1,20}] (* _Vaclav Kotesovec_, Mar 06 2014 *)

%t 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 *)

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

%o (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)}

%o for(n=1,20,print1(a(n),", "))

%o (Magma) [(&+[Binomial(n,k)*Binomial(n-1, k)^3: k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Apr 05 2021

%o (SageMath) [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

%Y Cf. A181067 (variant), A181068.

%Y Cf. A005260, A198256.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 08 2010