A136630 -id:A136630 - cp's OEIS frontend

A196776 Triangle T(n,k) gives the number of ordered partitions of an n set into k odd-sized blocks.

1, 0, 2, 1, 0, 6, 0, 8, 0, 24, 1, 0, 60, 0, 120, 0, 32, 0, 480, 0, 720, 1, 0, 546, 0, 4200, 0, 5040, 0, 128, 0, 8064, 0, 40320, 0, 40320, 1, 0, 4920, 0, 115920, 0, 423360, 0, 362880, 0, 512, 0, 130560, 0, 1693440, 0, 4838400, 0, 3628800

Offset: 1

Peter Bala, Oct 06 2011

See A136630 for the case of unordered partitions into odd-sized blocks. See A193474 for this triangle in row reverse form (but with an offset of 0).

			Triangle begins
.n\k.|..1....2....3....4.....5....6.....7
= = = = = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..0....2
..3..|..1....0....6
..4..|..0....8....0...24
..5..|..1....0...60....0...120
..6..|..0...32....0..480.....0..720
..7..|..1....0..546....0..4200....0..5040
...
T(4,2) = 8: The 8 ordered partitions of the set {1,2,3,4} into 2 odd-sized blocks are {1}{2,3,4}, {2,3,4}{1}, {2}{1,3,4}, {1,3,4}{2}, {3}{1,2,4}, {1,2,4}{3}, {4}{1,2,3} and {1,2,3}{4}.
Example of recurrence relation: T(7,3) = 3^2*T(5,3) + 3*(3-1)*T(5,1) = 9*60 + 6*1 = 546.

Cf. A006154 (row sums), A136630, A162590, A193474 (row reverse).

T(n,k) = 1/(2^k)*sum {j = 0..k}(-1)^(k-j)*binomial(k,j)*(2*j-k)^n.
Recurrence: T(n+2,k) = k^2*T(n,k) + k*(k-1)*T(n,k-2).
E.g.f.: x*sinh(t)/(1-x*sinh(t)) = x*t + 2*x^2*t^2/2! + (x+6*x^3)*t^3/3! + (8*x^2+24*x^4)*t^4/4! + (x+60*x^3+120*x^5)*t^5/5! + ....
O.g.f. for column 2*k: (2*k)!*x^(2*k)/Product {j = 0..k} (1 - (2*j)^2*x^2).
O.g.f. for column 2*k+1: (2*k+1)!*x^(2*k+1)/Product {j = 0..k} (1 - (2*j+1)^2*x^2).
Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of (I-t*M)^-1 (apart from the initial 1) lists the row polynomials for the present triangle.
n-th row sum = A006154(n).
Row generating polynomials equal D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A136630. - Peter Bala, Dec 06 2011

A144015 Expansion of e.g.f. 1/(1 - sin(4*x))^(1/4).

Original entry on oeis.org

1, 1, 5, 29, 265, 3001, 42125, 696149, 13296145, 287706481, 6959431445, 186061833869, 5448382252825, 173418192216361, 5961442393047965, 220112963745653189, 8687730877758518305, 365023930617143804641, 16266420334783460443685, 766297734521812843642109

Offset: 0

Paul D. Hanna, Sep 09 2008

nonn

Row sums of A186492 - Peter Bala, Feb 22 2011.

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 29*x^3/3! + 265*x^4/4! + 3001*x^5/5! +...
log(A(x)) = x + 4*x^2/2! + 16*x^3/3! + 128*x^4/4! + 1280*x^5/5! +...
A(x)^2/A(-x)^2 = 1 + 4*x + 16*x^2/2! + 128*x^3/3! +...+ 4^n*A000111(n)*x^n/n! +...
O.g.f.: 1/(1-x - 4*1*1*x^2/(1-5*x - 4*2*3*x^2/(1-9*x - 4*3*5*x^2/(1-13*x - 4*4*7*x^2/(1-17*x - 4*5*9*x^2/(1-...)))))) [continued fraction by Sergei Gladkovskii].

Cf. A000111, A001586, A007788, A186492, A230134, A227544, A230114, A235329.

Cf. A007696, A136630.

CoefficientList[Series[1/(1-Sin[4*x])^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(2*X)-sin(2*X))^(-1/2), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=exp(intformal(A^2/subst(A^2,x,-x))));n!*polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))

/* From A'(x) = A(x)^3 / A(-x)^2: */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^3/subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

/* 1/sqrt(1-2*Series_Reversion(Integral 1/sqrt(1+4*x-4*x^2) dx)): */
{a(n)=local(A=1);A=1/sqrt(1-2*serreverse(intformal(1/sqrt(1+4*x-4*x^2 +x*O(x^n)))));n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007696(n) = prod(k=0, n-1, 4*k+1);
a(n) = sum(k=0, n, a007696(k)*(4*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(2*x) - sin(2*x))^(-1/2).
(2) A(x)^2/A(-x)^2 = 1/cos(4*x) + tan(4*x).
(3) A(x) = exp( Integral A(x)^2/A(-x)^2 dx).
(4) A'(x) = A(x)^3/A(-x)^2 with A(0) = 1.
(5) A(x) = 1/sqrt(1 - 2*Series_Reversion( Integral 1/sqrt(1+4*x-4*x^2) dx )).
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013.
a(n) ~ 2^(3*n+5/4)*n^n/(exp(n)*Pi^(n+1/2)). - Vaclav Kotesovec, Jun 26 2013
a(n) = Sum_{k=0..n} A007696(k) * (4*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A160562 Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 91, 35, 1, 1, 820, 966, 84, 1, 1, 7381, 24970, 5082, 165, 1, 1, 66430, 631631, 273988, 18447, 286, 1, 1, 597871, 15857205, 14057043, 1768195, 53053, 455, 1, 1, 5380840, 397027996, 704652312, 157280838, 8187608, 129948, 680, 1

Offset: 0

Jonathan Vos Post, May 19 2009

This is table 4 on page 12 of Gelineau and Zeng, read downwards by columns.
Reversing rows gives A008958.
Apparently the table can also be obtained by deleting each second row and column of A136630.

			Triangle starts:
  1;
  1,     1;
  1,    10,      1;
  1,    91,     35,      1;
  1,   820,    966,     84,     1;
  1,  7381,  24970,   5082,   165,   1;
  1, 66430, 631631, 273988, 18447, 286, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)
Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022.
Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 18 2009.

Cf. A002452 (column k=1), A002453 (column k=2), A000447 (right column k=n-1), A185375 (right column k=n-2).

Cf. A001819, A008275, A008277, A008958, A039599, A136630.

A160562 := proc(n,k) npr := 2*n+1 ; kpr := 2*k+1 ; sinh(t*sinh(x)) ; npr!*coeftayl(%,x=0,npr) ; coeftayl(%,t=0,kpr) ; end: seq(seq(A160562(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Sep 09 2009

T[n_, k_] := Sum[(-1)^(k - m)*(2m + 1)^(2n + 1)*Binomial[2k, k + m]/(k + m + 1), {m, 0, k}]/(4^k*(2k)!);
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2017 *)

T(n,k) = (1/((2*k)!*4^k)) * Sum_{m=0..k} (-1)^(k-m)*A039599(k,m)*(2*m+1)^(2*n). - Werner Schulte, Nov 01 2015

T(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1) = ((2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sinh(x)^(2*k+1). Note that sin(x)^(2*k+1) = (Sum_{i=0..k} (-1)^i*binomial(2*k+1,k-i)*sin((2*i+1)*x))/(2^(2*k)). - Jianing Song, Oct 29 2023

More terms from R. J. Mathar, Sep 09 2009

A295254 Expansion of e.g.f. csch(x)(1 - sqrt(1 - 4sinh(x)))/2.

Original entry on oeis.org

1, 1, 4, 31, 352, 5341, 101824, 2341291, 63092992, 1950837241, 68093599744, 2648776394551, 113633946898432, 5330308817264341, 271416230974603264, 14910196369733535811, 879003840976919068672, 55354496206857969062641, 3708594029795800700944384, 263391744037123969891925071

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

nonn

Cf. A000108, A136630, A295237, A295255, A295256, A295257, A295258.

a:=series(csch(x)*(1-sqrt(1-4*sinh(x)))/2,x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019

nmax = 19; CoefficientList[Series[Csch[x] (1 - Sqrt[1 - 4 Sinh[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Sinh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2) * 17^(1/4) * n^(n-1) / (exp(n) * (log((1+ sqrt(17))/4))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} k! * binomial(2*k+1,k)/(2*k+1) * A136630(n,k). - Seiichi Manyama, Feb 23 2025

A185690 Exponential Riordan array (1,sin(x)).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -4, 0, 1, 1, 0, -10, 0, 1, 0, 16, 0, -20, 0, 1, -1, 0, 91, 0, -35, 0, 1, 0, -64, 0, 336, 0, -56, 0, 1, 1, 0, -820, 0, 966, 0, -84, 0, 1, 0, 256, 0, -5440, 0, 2352, 0, -120, 0, 1, -1, 0, 7381, 0, -24970, 0, 5082, 0, -165, 0, 1, 0, -1024, 0, 87296, 0, -90112, 0, 10032, 0, -220, 0, 1

Offset: 1

Vladimir Kruchinin, Feb 10 2011

The row n=0 with T(0,0)=1 and the column T(n,0)=0, n>0, are not entered into the sequence here.
A signed version of A136630 (apart from row 0 and column 0). - Peter Bala, Oct 06 2011
Also the Bell transform of the sequence "a(n) = (-1)^(n/2) if n is even else 0" without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Array begins:
   1;
   0,   1;
  -1,   0,   1;
   0,  -4,   0,   1;
   1,   0, -10,   0,   1;
   0,  16,   0, -20,   0,   1;
  -1,   0,  91,   0, -35,   0,   1;
   0, -64,   0, 336,   0, -56,   0,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565, [math.CO], 2010.

Cf. A000111, A136630.

A185690 := proc(n,k) if type(k+n,'even') then 2^(1-k)/k! * add( (-1)^(floor((n+k)/2)-i)*binomial(k,i)*(2*i-k)^n,i=0..floor(k/2)) ; else 0; end if; end proc: # R. J. Mathar, Feb 21 2011
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::even,(-1)^(n/2),0), 10); # Peter Luschny, Jan 29 2016

t[n_, k_] /; OddQ[n - k] = 0; t[n_, k_] /; EvenQ[n - k] := 2^(1-k)/k!* Sum[ (-1)^(Floor[(n+k)/2] - i)*Binomial[k, i]*(2*i-k)^n, {i, 0, k/2}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, If[EvenQ[n], (-1)^(n/2), 0]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

from sympy import binomial, factorial as f, floor
def T(n, k):
    return 0 if (n - k)%2 else 2**(1 - k)*sum([(-1)**((n + k)//2 - i)*binomial(k, i)*(2*i - k)**n for i in range(k//2 + 1)])//f(k)
for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jul 11 2017

T(n,k) = 2^(1-k)/k! *Sum_{i=0..floor(k/2)} (-1)^(floor((n+k)/2)-i) *binomial(k,i) *(2*i-k)^n, for even(n-k).

Sum_{k=0..n} T(n+1,k+1)*k! = A000111(n). - Alexander Burstein, Aug 01 2025

A227544 Expansion of e.g.f. 1/(1 - sin(6*x))^(1/6).

Original entry on oeis.org

1, 1, 7, 55, 721, 11761, 240247, 5801095, 162512161, 5171130721, 184337942887, 7275081518935, 314918762166001, 14834964193292881, 755507853144691927, 41362173671901329575, 2422478811455080626241, 151132171549872325122241, 10006051653759338150151367, 700695219796759105368529015

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - Vaclav Kotesovec, Jan 03 2014

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 55*x^3/3! + 721*x^4/4! + 11761*x^5/5! + ...
where A(x)^3 = 1 + 3*x + 27*x^2/2! + 297*x^3/3! + 4617*x^4/4! + 87723*x^5/5! + ...
and 1/A(x)^3 = 1 - 3*x - 9*x^2/2! + 27*x^3/3! + 81*x^4/4! - 243*x^5/5! + ...
which illustrates 1/A(x)^3 = cos(3*x) - sin(3*x).
O.g.f.: 1/(1-x - 6*1*1*x^2/(1-7*x - 6*2*4*x^2/(1-13*x - 6*3*7*x^2/(1-19*x - 6*4*10*x^2/(1-25*x - 6*5*13*x^2/(1-...)))))), a continued fraction.

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A235128 (p=7), A230114 (p=8).

Cf. A008542, A136630.

CoefficientList[Series[1/(1-Sin[6*x])^(1/6), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(3*X)-sin(3*X))^(-1/3), n)}
for(n=0,20,print1(a(n),", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^3/subst(A^3, x, -x)))); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a008542(n) = prod(k=0, n-1, 6*k+1);
a(n) = sum(k=0, n, a008542(k)*(6*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(3*x) - sin(3*x))^(-1/3).
(2) A(x)^3/A(-x)^3 = 1/cos(6*x) + tan(6*x).
(3) A(x) = exp( Integral A(x)^3/A(-x)^3 dx ).
O.g.f.: 1/G(0) where G(k) = 1 - (6*k+1)*x - 6*(k+1)*(3*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * 2^(2*n+1/2) * 3^n / (Gamma(1/3) * n^(2/3) * Pi^(n+1/3)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A008542(k) * (6*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A230114 Expansion of e.g.f. 1/(1 - sin(8*x))^(1/8).

Original entry on oeis.org

1, 1, 9, 89, 1521, 32401, 869049, 27608489, 1019581281, 42824944801, 2017329504489, 105299243488889, 6032850630082641, 376363074361201201, 25396689469918450329, 1843101478742259481289, 143145930384321475601601, 11846611289341729822881601, 1040750126963789832859930569

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) is a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - Vaclav Kotesovec, Jan 03 2014

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 89*x^3/3! + 1521*x^4/4! + 32401*x^5/5! + ...
where A(x)^4 = 1 + 4*x + 48*x^2/2! + 704*x^3/3! + 14592*x^4/4! + 369664*x^5/5! + ...
and 1/A(x)^4 = 1 - 4*x - 16*x^2/2! + 64*x^3/3! + 256*x^4/4! - 1024*x^5/5! + ...
which illustrates 1/A(x)^4 = cos(4*x) - sin(4*x).
O.g.f.: 1/(1-x - 8*1*1*x^2/(1-9*x - 8*2*5*x^2/(1-17*x - 8*3*9*x^2/(1-25*x - 8*4*13*x^2/(1-33*x - 8*5*17*x^2/(1-...)))))), a continued fraction.

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A235128 (p=7).

Cf. A045755, A136630.

CoefficientList[Series[1/(1-Sin[8*x])^(1/8), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(4*X)-sin(4*X))^(-1/4), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^4/subst(A^4, x, -x)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045755(n) = prod(k=0, n-1, 8*k+1);
a(n) = sum(k=0, n, a045755(k)*(8*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(4*x) - sin(4*x))^(-1/4).
(2) A(x)^4/A(-x)^4 = 1/cos(8*x) + tan(8*x).
(3) A(x) = exp( Integral A(x)^4/A(-x)^4 dx ).
O.g.f.: 1/G(0) where G(k) = 1 - (8*k+1)*x - 8*(k+1)*(4*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * 2^(4*n+3/8) / (Gamma(1/4) * n^(3/4) * Pi^(n+1/4)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A045755(k) * (8*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A230134 Expansion of e.g.f. 1/(1 - sin(5*x))^(1/5).

Original entry on oeis.org

1, 1, 6, 41, 456, 6301, 108576, 2207981, 52012416, 1390239481, 41593598976, 1376769180401, 49955931795456, 1971671764875541, 84095262825824256, 3854514200269774901, 188942180401957502976, 9863099585213327293681, 546266997049408050364416, 31993839349571172423492281

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 41*x^3/3! + 456*x^4/4! + 6301*x^5/5! +...
O.g.f.: 1/(1-x - 5*1*2/2*x^2/(1-6*x - 5*2*7/2*x^2/(1-11*x - 5*3*12/2*x^2/(1-16*x - 5*4*17/2*x^2/(1-21*x - 5*5*22/2*x^2/(1-...)))))), a continued fraction.

Cf. A001586, A007788, A144015, A227544, A230114.

Cf. A008548, A136630.

CoefficientList[Series[1/(1-Sin[5*x])^(1/5), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((1-sin(5*X))^(-1/5), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^(5/2)/subst(A^(5/2), x, -x)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a008548(n) = prod(k=0, n-1, 5*k+1);
a(n) = sum(k=0, n, a008548(k)*(5*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies: A(x) = (cos(5*x/2) - sin(5*x/2))^(-2/5).
O.g.f.: 1/G(0) where G(k) = 1 - (5*k+1)*x - 5*(k+1)*(5*k+2)/2*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * sqrt(5+sqrt(5)) * Gamma(3/5) * 2^(n-9/10) * 5^n / (n^(3/5) * Pi^(n+7/5)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A008548(k) * (5*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A107403 Expansion of e.g.f. 1/(1 - 3*sinh(x)).

Original entry on oeis.org

1, 3, 18, 165, 2016, 30783, 564048, 12057825, 294587136, 8096756763, 247266851328, 8306410495485, 304403359942656, 12085026305182743, 516690458532292608, 23668814542820609145, 1156515067746149400576, 60041982382475841900723, 3300519734382436473765888

Offset: 0

Miklos Kristof, Jun 09 2005

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..379

Cf. A000557, A006154, A136630.

E(x):=1/(1-3*sinh(x)): f[0]:=E(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..30);

CoefficientList[Series[1/(1-3*Sinh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

a(n) ~ n!/(sqrt(10)*(log(1/3+sqrt(10)/3))^(n+1)). - Vaclav Kotesovec, Jun 26 2013
a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
a(n) = Sum_{k=0..n} 3^k * k! * A136630(n,k). - Seiichi Manyama, Jun 25 2025

A121408 Triangle T(n,k) defined by the generating function: exp(yarcsin(x))-1 = Sum_{n>=1} (Sum_{k=1..n} T(n,k)y^k)*x^n/n!.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 9, 0, 10, 0, 1, 0, 64, 0, 20, 0, 1, 225, 0, 259, 0, 35, 0, 1, 0, 2304, 0, 784, 0, 56, 0, 1, 11025, 0, 12916, 0, 1974, 0, 84, 0, 1, 0, 147456, 0, 52480, 0, 4368, 0, 120, 0, 1, 893025, 0, 1057221, 0, 172810, 0, 8778, 0, 165, 0, 1, 0, 14745600, 0

Offset: 1

Emeric Deutsch, Jul 28 2006

Row sums are equal to A006228(n). This is sequence A091885 with additional intertwining zeros.
F(n,m) = n!*T(n,m)/m! is a composite (akin to Riordan arrays) of F(x)=arcsin(x) and (F(x))^m = Sum_{n>=m} F(n,m)*x^n, and for o.g.f. G(x), G(arcsin(x)) = g(0) +Sum_{n>=1} Sum_{m=1..n} F(n,m)*g(m)*x^n, see the preprint. - Vladimir Kruchinin, Feb 10 2011
The unsigned matrix inverse is A136630 (with a different offset). - Peter Bala, Feb 23 2011
Also the Bell transform of A177145. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle starts:
  1;
  0,1;
  1,0,1;
  0,4,0,1;
  9,0,10,0,1;
  0,64,0,20,0,1;
Row polynomials R(6,x) = x^2*(x^2 + 2^2)*(x^2 + 4^2) = 64*x^2 + 20*x^4 + x^6 and
R(7,x) = x*(x^2 + 1)*(x^2 + 3^2)*(x^2 + 5^2) = 225*x + 259*x^3 + 35*x^5 + x^7. - _Peter Bala_, Aug 29 2012

B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A000111, A000142, A006228, A091885, A136630. A182971.

g:=exp(y*arcsin(x))-1: gser:=simplify(series(g,x=0,15)): for n from 1 to 12 do P[n]:=sort(n!*coeff(gser,x,n)) od: for n from 1 to 12 do seq(coeff(P[n],y,k),k=1..n) od; # yields sequence in triangular form
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::odd,0,doublefactorial(n-1)^2), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[OddQ[#], 0, (# - 1)!!^2] &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

T(n,m) = ((n-1)!/(m-1)!) *sum_{k=1..n-m} sum_{j=1..k} binomial(k,j) *(2^(1-j) /(n-m+j)!) *sum{i=0..floor(j/2)} (-1)^((n-m)/2-i-j) *binomial(j,i) *(j-2*i)^(n-m+j) *binomial(k+n-1,n-1), n>m and even(n-m). [Vladimir Kruchinin, Feb 10 2011]
From Peter Bala, Aug 29 2012: (Start)
See A182971 for a version of the row reverse of this triangle.
Even-indexed row polynomial R(2*n,x) = x^2*prod(k=1..n-1, (x^2 + (2*k)^2) ).
Odd-indexed row polynomial R(2*n+1,x) = x*prod(k=1..n, (x^2 + (2*k-1)^2) ). See Berndt p.263. (End)
Sum_{k=0..n} T(n+1,k+1)*A000111(k) = n! = A000142(n). - Alexander Burstein, Aug 01 2025

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