A039619 Second column of Jabotinsky-triangle A038455 related to A006963.

1, 9, 107, 1650, 31594, 725592, 19471500, 598482000, 20742534576, 800575997760, 34059828307680, 1583808130195200, 79925022369273600, 4350478314951982080, 254086498336122950400, 15849890120755311667200

Offset

2,2

Comments

Explicit formula using partitions of n into 2 parts: cf. Knuth's paper for f(n,2), n >= 2, formula with f(k) as given above.

Links

Table of n, a(n) for n=2..17.

D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.

D. E. Knuth, Problem 11832,The American Mathematical Monthly, Vol. 122, No. 4 (April 2015), page 390.

David Callan, An identity for A039619.

Formula

a(n) = sum(binomial(n-1, j-1)*f(j)*f(n-j), j=1..n-1) with f(k) := A006963(k+1) = (2*k+1)!/k!, k >= 1.

E.g.f.: log((1-sqrt(1-4*x))/x/2)^2/2. - Vladeta Jovovic, May 02 2003

a(n) ~ log(2) * n^(n-1) * 2^(2*n-1/2) / exp(n). - Vaclav Kotesovec, Oct 07 2013

a(n) = (H(2*n-1) - H(n)) * (2*n-1)! / n! where H() is Harmonic numbers [Knuth 2015]. - Michael Somos, May 11 2015

a(n) = (n-1)!*sum(binomial(2n-1-k, n-1-k)/k,k=1..n-1) [Callan 2015]. - David Callan, May 17 2015

Example

G.f. = x^2 + 9*x^3 + 107*x^4 + 1650*x^5 + 31594*x^6 + 725592*x^7 + ...

Mathematica

Rest[Rest[CoefficientList[Series[Log[(1-Sqrt[1-4*x])/x/2]^2/2, {x, 0, 20}], x]* Range[0, 20]!]] (* Vaclav Kotesovec, Oct 07 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ 1/k, {k, n + 1, 2 n - 1}] (2 n - 1)!/n!]; (* Michael Somos, May 11 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ Times @@ Drop[ Range[ n - 1] + n, {k}], {k, n - 1}]]; (* Michael Somos, May 11 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sum(k=n+1, 2*n-1, 1/k) * (2*n-1)! / n!)}; /* Michael Somos, May 11 2015 */

Crossrefs

Cf. A006963, A038455.

Cf. A039646.

Sequence in context: A367789 A316145 A365032 * A249048 A080505 A104224

Adjacent sequences: A039616 A039617 A039618 * A039620 A039621 A039622

Keyword

nonn

Author

Wolfdieter Lang

Status

approved