A166990 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

1, 2, 7, 30, 147, 786, 4472, 26644, 164477, 1044258, 6782484, 44887236, 301782361, 2056250570, 14172792355, 98667874038, 692948001906, 4904403499992, 34951124337300, 250617829087656, 1807055528439771, 13095146839953030

Offset

0,2

Comments

Analogous to the square of the g.f. of Catalan numbers (A000108):

C(x)^2 = exp( Sum_{n>=1} A000984(n)*x^n/n ) where central binomial coefficient A000984(n) = Sum_{k=0..n} C(n,k)^2.

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Formula

Self-convolution of A166991.

a(n) ~ c * 8^n / n^2, where c = 0.58462... - Vaclav Kotesovec, Nov 27 2017

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +...
log(A(x)) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...

Mathematica

a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/n, {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3)*x^m/m)+x*O(x^n)), n)}

Crossrefs

Cf. A000172 (Franel numbers), A166991, A166992, A218117, A218119.

Sequence in context: A243632 A196148 A193464 * A059578 A136574 A290875

Adjacent sequences: A166987 A166988 A166989 * A166991 A166992 A166993

Keyword

nonn

Author

Paul D. Hanna, Nov 17 2009

Status

approved