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A181067 a(n) = Sum_{k=0..n-1} C(n-1,k)^2 * C(n,k). 3
1, 3, 16, 95, 606, 4032, 27616, 193167, 1372930, 9881498, 71846160, 526764680, 3889340560, 28888634400, 215680108416, 1617467908751, 12177754012458, 92004463332486, 697263463622080, 5298985086555090, 40371796982444356 (list; graph; refs; listen; history; text; internal format)
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} C(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

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 + ...

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 */

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

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Last modified October 19 15:50 EDT 2019. Contains 328223 sequences. (Running on oeis4.)