A125054 Central terms of triangle A125053.

1, 3, 21, 327, 9129, 396363, 24615741, 2068052367, 225742096209, 31048132997523, 5252064083753061, 1071525520294178007, 259439870666594250489, 73542221109962636293083, 24125551094579137082039181, 9068240688454120376775401247, 3871645204706420218816959159969

Offset

0,2

Comments

Triangle A125053 is a variant of triangle A008301 (enumeration of binary trees) such that the leftmost column is the secant numbers (A000364).

Right edge of triangle A210108.

Apparently all terms (except the initial 1) have 3-valuation 1. - F. Chapoton, Aug 02 2021

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)

Formula

Binomial transform of A000182 (e.g.f. tan(x)).

a(n) = Sum_{k=0..n} A130847(n,k)*2^k. - Philippe Deléham, Jul 22 2007

G.f.: 1/(1-sqrt(x))/Q(0), where Q(k)= 1 + sqrt(x) - x*(2*k+1)*(2*k+2)/(1 - sqrt(x) - x*(2*k+2)*(2*k+3)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 27 2013

G.f.: Q(0)/(1-3*x), where Q(k) = 1 - 4*x^2*(2*k+1)*(2*k+3)*(k+1)^2/( 4*x^2*(2*k+1)*(2*k+3)*(k+1)^2 - (1 - 8*x*k^2 - 8*x*k -3*x)*(1 - 8*x*k^2 - 24*x*k -19*x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2013

G.f.: Q(0)/(1-1*x), where Q(k) = 1 - (2*k+1)*(2*k+2)*x/(x*(2*k+1)*(2*k+2) - (1-x)/(1 - (2*k+2)*(2*k+3)*x/(x*(2*k+2)*(2*k+3) - (1-x)/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

a(n) ~ 2^(4*n+5) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 30 2015

From Peter Bala, May 11 2017: (Start)

O.g.f. as an S-fraction: A(x) = 1/(1 - 3*x/(1 - 4*x/(1 - 15*x/(1 - 16*x/(1 - 35*x/(1 - 36*x/(1 - ...))))))), where the unsigned coefficients in the partial numerators [3, 4, 15, 16, 35, 36, ...] come in pairs of the form 4*n^2 - 1, 4*n^2 for n = 1,2,....

A(x) = 1/(1 + 3*x - 6*x/(1 - 2*x/(1 + 3*x - 20*x/(1 - 12*x/(1 + 3*x - 42*x/(1 - 30*x/(1 + 3*x - ...))))))), , where the unsigned coefficients in the partial numerators [6, 2, 20, 12, 42, 30, ...] are obtained from the sequence [2, 6, 12, 20, ..., n*(n + 1), ...] by swapping adjacent terms. (End)

Mathematica

b[n_]:=n!*SeriesCoefficient[Tan[x], {x, 0, n}]; Table[Sum[Binomial[n, k]*b[2*k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 30 2015 *)

Crossrefs

Cf. A125053, A125055, A000182.

Sequence in context: A134528 A332974 A118410 * A113085 A240936 A342245

Adjacent sequences: A125051 A125052 A125053 * A125055 A125056 A125057

Keyword

nonn

Author

Paul D. Hanna, Nov 21 2006

Status

approved