A130847 -id:A130847 - cp's OEIS frontend

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

Paul D. Hanna, Nov 21 2006

nonn

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

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)

Cf. A125053, A125055, A000182.

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 *)

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)

A126152 Main diagonal of symmetric triangle A126150: a(n) = A126150(n,n).

Original entry on oeis.org

1, 4, 36, 744, 28536, 1736064, 152914176, 18372559104, 2885671339776, 573765893121024, 140835811776316416, 41820352964911908864, 14774712204104658671616, 6124078747943873540112384

Offset: 0

Paul D. Hanna, Dec 19 2006

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..232

Cf. A126150; A126151 (column 0), A126153 (diagonal).

/* Continued fraction involving even-indexed pentagonal numbers: */
{a(n)=local(CF=1+x*O(x),m,P); for(k=1, n,m=2*((n-k)\2+1);P=m*(3*m-1)/2-((n-k+1)%2); CF=1/(1-P*x*CF)); polcoeff(CF, n, x)}
for(n=0,20,print1(a(n),","))

a(n) = Sum_{k=0..n} A130847(n,k)*3^k. - Philippe Deléham, Jul 22 2007
G.f.: 1/(1 - 4*x/(1 - 5*x/(1 - 21*x/(1 - 22*x/(1 - 50*x/(1 - 51*x/(1 - 91*x/(1 - 92*x/(1 -...))))))))), a continued fraction involving even-indexed pentagonal numbers A000326. - Paul D. Hanna, Feb 15 2012
a(n) ~ Gamma(1/3) * 2^(3*n+7/3) * 3^(n+3/2) * n^(2*n+7/6) / (exp(2*n) * Pi^(2*n+13/6)). - Vaclav Kotesovec, May 30 2015

A126157 Main diagonal and central terms of symmetric triangle A126155.

Original entry on oeis.org

1, 5, 55, 1415, 69025, 5403005, 616437655, 96365988815, 19756766836225, 5140259013390005, 1654190282113104055, 645005323804145184215, 299613583126435893179425, 163464903833408195554809005

Offset: 0

Paul D. Hanna, Dec 20 2006

nonn

Cf. A126155; A126156 (column 0), A126158 (diagonal).

a(n)=Sum_{k, 0<=k<=n}A130847(n,k)*4^k. - Philippe Deléham, Jul 22 2007

cp's OEIS Frontend

A125054 Central terms of triangle A125053.

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A126152 Main diagonal of symmetric triangle A126150: a(n) = A126150(n,n).

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A126157 Main diagonal and central terms of symmetric triangle A126155.

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