A181067 a(n) = Sum_{k=0..n-1} binomial(n-1,k)^2 * binomial(n,k).

1, 3, 16, 95, 606, 4032, 27616, 193167, 1372930, 9881498, 71846160, 526764680, 3889340560, 28888634400, 215680108416, 1617467908751, 12177754012458, 92004463332486, 697263463622080, 5298985086555090, 40371796982444356

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Formula

L.g.f.: Sum_{n>=1} [ Sum_{k>=0} binomial(n+k-1,k)^3 *x^k ] *x^n/n.

Logarithmic derivative of A181066.

Recurrence: n^2*a(n) = - (n^2-17*n+10)*a(n-1) + 48*(n^2-3*n+1)*a(n-2) + 16*(n-3)*(11*n-36)*a(n-3) + 128*(n-4)^2*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ sqrt(3)*8^n/(6*Pi*n). - Vaclav Kotesovec, Oct 24 2012

a(n) = 3F2([1-n, 1-n, -n], [1, 1], -1). - Pierre-Louis Giscard, Jul 20 2013

a(n) = n * hypergeometric([-n+1,-n+1,-n+1], [1,2], -1) for n > 0. - Emanuele Munarini, Sep 27 2016

a(n) = Sum_{k=0..n-1} ((n-k)/n)^2 * binomial(n,k)^3. - G. C. Greubel, Apr 05 2021

Example

L.g.f.: L(x) = x + 3*x^2/2 + 16*x^3/3 + 95*x^4/4 + 606*x^5/5 + ...
which equals the series:
  L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...)*x
  + (1 + 2^3*x +  3^3*x^2 +  4^3*x^3 +   5^3*x^4 +   6^3*x^5 + ...)*x^2/2
  + (1 + 3^3*x +  6^3*x^2 + 10^3*x^3 +  15^3*x^4 +  21^3*x^5 + ...)*x^3/3
  + (1 + 4^3*x + 10^3*x^2 + 20^3*x^3 +  35^3*x^4 +  56^3*x^5 + ...)*x^4/4
  + (1 + 5^3*x + 15^3*x^2 + 35^3*x^3 +  70^3*x^4 + 126^3*x^5 + ...)*x^5/5
  + (1 + 6^3*x + 21^3*x^2 + 56^3*x^3 + 126^3*x^4 + 252^3*x^5 + ...)*x^6/6 + ...
Exponentiation yields the g.f. of A181066:
  exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 157*x^5 + 865*x^6 + ... + A181066(n)*x^n + ...

Maple

A181067:= n-> add(((n-k)/n)^2*binomial(n, k)^3, k=0..n-1); seq(A181067(n), n=1..25); # G. C. Greubel, Apr 05 2021

Mathematica

Table[Sum[Binomial[n-1, k]^2*Binomial[n, k], {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 24 2012 *)
Table[HypergeometricPFQ[{1-n, 1-n, -n}, {1, 1}, -1], {n, 1, 20}] (* Pierre-Louis Giscard, Jul 20 2013 *)

Prog

(PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^3*n/(n-k))}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1, k)^3*x^k)*x^m/m)+x*O(x^n), n)}
(Maxima) makelist(hypergeometric([-n+1, -n+1, -n], [1, 1], -1), n, 0, 12); /* Emanuele Munarini, Sep 27 2016 */
(Magma) [(&+[ ((n-k)/n)^2*Binomial(n, k)^3 : k in [0..n-1]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021
(Sage) [sum( ((n-k)/n)^2*binomial(n, k)^3 for k in (0..n-1) ) for n in (1..25)] # G. C. Greubel, Apr 05 2021

Crossrefs

Cf. A181066 (exp), A181069 (variant).

Sequence in context: A074555 A137644 A114174 * A006347 A000270 A157051

Adjacent sequences: A181064 A181065 A181066 * A181068 A181069 A181070

Keyword

nonn

Author

Paul D. Hanna, Oct 03 2010

Status

approved