A321799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^5).

1, 1, 6, 31, 176, 1071, 6797, 44493, 298279, 2037550, 14131441, 99244564, 704360703, 5043969503, 36399930179, 264451303466, 1932650461883, 14198082537190, 104792195449688, 776681663951998, 5778226417888171, 43135097969972931, 323012620411650708, 2425745980876575899, 18264470545275495152

Offset: 0

Author

Ludovic Schwob, Nov 19 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..199 from Ludovic Schwob)

Crossrefs

Cf. A109081, A161797, A321798.

Programs

GAP

List([0..25],n->Sum([0..n],k->Binomial(n,k)/(n-k+1)*Binomial(n+4*k-1,n-k))); # Muniru A Asiru, Nov 24 2018

Magma

[1] cat [&+[(Binomial(n,k)/(n-k+1)) * Binomial(n+4*k-1,n-k):  k in [0..n]]: n in [1.. 25]]; // Vincenzo Librandi, Dec 08 2018

Maple

eq:= a - 1/(1-x/(1-x*a)^5):
S:= series(RootOf(numer(eq),a),x,31):
seq(coeff(S,x,j),j=0..30); # Robert Israel, Dec 10 2018

Mathematica

a[n_]:=Sum[ Binomial[n,k]/(n-k+1)*Binomial[n+4*k-1,n-k], {k,0,n}]; Array[a, 20, 0] (* Stefano Spezia, Nov 19 2018 *)
A[] = 0; Do[A[x] = 1/(1-x/(1-x*A[x])^5)+O[x]^25, {25}];
CoefficientList[A[x], x] (* Jean-François Alcover, Dec 30 2018 *)

PARI

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+4*k-1, n-k)/(n-k+1)); \\ Michel Marcus, Nov 19 2018

Sage

[sum(binomial(n,k)*binomial(n+4*k-1,n-k)/(n-k+1) for k in (0..n)) for n in range(25)] # G. C. Greubel, Dec 14 2018

Formula

a(n) = Sum_{k=0..n} (C(n,k)/(n-k+1)) * C(n+4*k-1,n-k).

a(n) ~ sqrt((1 - r*s)*(1 + 4*r*s) / (5*Pi*(5*s - 2))) / (2 * n^(3/2) * r^(n+1)), where r = 0.124910212976238209867004924637837518925706044646... and s = 1.72708330560542094133450070142549940430523638921... are real roots of the system of equations s*(1 - r/(1 - r*s)^5) = 1, 5*r^2*s^2 = (1 - r*s)^6. - Vaclav Kotesovec, Nov 21 2018