A006228 -id:A006228 - cp's OEIS frontend

A385343 Exponential Riordan array (1, arcsin(x)).

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

Offset: 0

Seiichi Manyama, Jun 26 2025

			Triangle starts:
  1;
  0,   1;
  0,   0,  1;
  0,   1,  0,   1;
  0,   0,  4,   0,  1;
  0,   9,  0,  10,  0,  1;
  0,   0, 64,   0, 20,  0, 1;
  0, 225,  0, 259,  0, 35, 0, 1;

Essentialy same as A121408.
Row sums give A006228.
Cf. A091885, A136630, A185951.

T(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);

E.g.f. of column k (with leading zeros): arcsin(x)^k / k!

T(n,k) = A121408(n,k) for k > 0.

A002019 a(n) = a(n-1) - (n-1)(n-2)a(n-2).

Original entry on oeis.org

1, 1, 1, -1, -7, 5, 145, -5, -6095, -5815, 433025, 956375, -46676375, -172917875, 7108596625, 38579649875, -1454225641375, -10713341611375, 384836032842625, 3663118565923375, -127950804666254375, -1519935859717136875

Offset: 0

N. J. A. Sloane

Dwight, Tables of Integrals ..., Eq. 552.5, page 133.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
G. Guillotte and L. Carlitz, Problem H-216 and solution, Fib. Quarter. p. 90, Vol 13, 1, Feb. 1975.
R. Kelisky, The numbers generated by exp(arctan x), Duke Math. J., 26 (1959), 569-581.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Bisections are A102058 and A102059.
Cf. A006228.
Row sums of signed triangle A049218.
Cf. A000246.

a002019 n = a002019_list !! n
a002019_list = 1 : 1 : zipWith (-)
   (tail a002019_list) (zipWith (*) a002019_list a002378_list)
-- Reinhard Zumkeller, Feb 27 2012

I:=[1,1]; [1] cat [ n le 2 select I[n] else Self(n-1)-(n^2-3*n+2)*Self(n-2): n in [1..35]]; // Vincenzo Librandi, May 02 2015

RecurrenceTable[{a[0]==1,a[1]==1,a[n]==a[n-1]-(n-1)(n-2)a[n-2]}, a[n],{n,30}] (* Harvey P. Dale, May 02 2011 *)
CoefficientList[Series[E^(ArcTan[x]),{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Nov 06 2014 *)

a(n):=n!*sum(if oddp(m+n) then 0 else (-1)^((3*n+m)/2)/(2^m*m!)*sum(2^i*binomial(n-1,i-1)*m!/i!*stirling1(i,m),i,m,n),m,1,n); /* Vladimir Kruchinin, Aug 05 2010 */

E.g.f.: exp(arctan(x)).
a(n) = n!*sum(if oddp(m+n) then 0 else (-1)^((3*n+m)/2)/(2^m*m!)*sum(2^i*binomial(n-1,i-1)*m!/i!*stirling1(i,m),i,m,n),m,1,n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: exp(arctan(x)) = 1 + 2x/(H(0)-x); H(k) = 4k + 2 + x^2*(4k^2 + 8k + 5)/H(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
a(n+1) = a(n) - a(n-1) * A002378(n-2). - Reinhard Zumkeller, Feb 27 2012
E.g.f.: -2i*(B((1+ix)/2; (2-i)/2, (2+i)/2) - B(1/2; (2-i)/2, (2+i)/2)), for a(0)=0, a(1)=a(2)=a(3)=1, B(x;a,b) is the incomplete Beta function. - G. C. Greubel, May 01 2015
a(n) = i^n*n!*Sum_{r+s=n} (-1)^s*binomial(-i/2, r)*binomial(i/2,s) where i is the imaginary unit. See the Fib. Quart. link. - Michel Marcus, Jan 22 2017

More terms from Herman P. Robinson

A101928 E.g.f. cos(arcsinh(x)) = sin(arccosh(x)) (even powers only).

Original entry on oeis.org

1, -1, 5, -85, 3145, -204425, 20646925, -2993804125, 589779412625, -151573309044625, 49261325439503125, -19753791501240753125, 9580588878101765265625, -5527999782664718558265625, 3742455852864014463945828125, -2937827844498251354197475078125, 2646982887892924470131925045390625

Offset: 1

Ralf Stephan, Dec 28 2004

sign

Absolute values are expansion of e.g.f. cosh(arcsin(x)).

			cos(arcsinh(x)) = 1 - x^2/2 + 5x^4/4! - 85x^6/6! + 3145x^8/8! - ...

Muniru A Asiru, Table of n, a(n) for n = 1..100

Bisection of A006228.

Cf. A079484, A277354.

seq(coeff(series(factorial(n)*cos(arcsinh(x)), x,n+1),x,n),n=0..40,2); # Muniru A Asiru, Jul 22 2018

Table[n!*SeriesCoefficient[Cos[ArcSinh[x]],{x,0,n}],{n,0,40,2}] (* Vaclav Kotesovec, Oct 23 2013 *)
Flatten[{1, Table[(-1)^(n+1)*Product[4*k^2 + 1, {k, 1, n}], {n, 0, 12}]}] (* Vaclav Kotesovec, Oct 10 2016 *)

E.g.f.: cos(arcsinh(x)) =  sqrt(1+x^2)*(1-x^2*(1-5*x^2/(G(0)+5*x^2))); G(k) = (k+2)*(2*k+3)-x^2*(2*k^2+6*k+5)+x^2*(k+2)*(2*k+3)*(2*k^2+10*k+13)/G(k+1);
For cosh(arcsin(x)) = sqrt(1-x^2)*(1 + x^2*(1 + 5*x^2/(G(0) - 5*x^2))); G(k) = x^2*(2*k^2+6*k+5) + (k+2)*(2*k+3) - x^2*(k+2)*(2*k+3)*(2*k^2+10*k+13)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 19 2011
G.f.: 1 - x*(1 + x*(G(0) - 1)/(x-1)) where G(k) = 1 + ((2*k+2)^2+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) ~ (-1)^(n+1) * sinh(Pi/2) * 2^(2*n-2) * n^(2*n-3) / exp(2*n). - Vaclav Kotesovec, Oct 23 2013
For n>1, a(n) = (-1)^(n+1) * A277354(n-2). - Vaclav Kotesovec, Oct 10 2016

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

A385369 Expansion of e.g.f. x + sqrt(x^2 + 1).

Original entry on oeis.org

1, 1, 1, 0, -3, 0, 45, 0, -1575, 0, 99225, 0, -9823275, 0, 1404728325, 0, -273922023375, 0, 69850115960625, 0, -22561587455281875, 0, 9002073394657468125, 0, -4348001449619557104375, 0, 2500100833531245335015625, 0, -1687568062633590601135546875, 0

Offset: 0

Seiichi Manyama, Jun 26 2025

Cf. A006228, A177698, A385343.

my(N=30, x='x+O('x^N)); Vec(serlaplace(x+sqrt(x^2+1)))

E.g.f.: exp(arcsinh(x)).
E.g.f. A(x) satisfies A(x) = 1/A(-x).
a(n) = Sum_{k=0..n} i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) = A177698(n-1) for n > 1.
a(2*n+1) = 0 for n > 0.
a(n) = 2^n * n! * binomial((n+1)/2,n)/(n+1).

A101927 E.g.f. of sin(arcsinh(x)) (odd powers only).

Original entry on oeis.org

1, -2, 20, -520, 26000, -2132000, 260104000, -44217680000, 9993195680000, -2898026747200000, 1049085682486400000, -463695871658988800000, 245758811979264064000000, -153845016299019304064000000, 112306861898284091966720000000, -94562377718355205435978240000000, 90969007365057707629411066880000000

Offset: 1

Ralf Stephan, Dec 28 2004

sign

Absolute values are expansion of sinh(arcsin(x)).

			sin(arcsinh(x)) = x - 2x^3/3! + 20x^5/5! - 520x^7/7! + 26000x^9/9! - ...

Muniru A Asiru, Table of n, a(n) for n = 1..100

Bisection of A006228.

seq(coeff(series(factorial(n)*sin(arcsinh(x)), x,n+1),x,n),n=1..30,2); # Muniru A Asiru, Jul 22 2018

Table[n!*SeriesCoefficient[Sin[ArcSinh[x]],{x,0,n}],{n,1,40,2}] (* Vaclav Kotesovec, Oct 23 2013 *)

E.g.f.: sin(arcsinh(x)) = x*sqrt(1+x^2)*(1 - 5*x^2/(G(0) + 5*x^2)); G(k) = (2*k+2)*(2*k+3) - x^2*(4*k^2+8*k+5) + x^2*(2*k+2)*(2*k+3)*(4*k^2+16*k+17)/G(k+1);
for sinh(arcsin(x)) = x*sqrt(1-x^2)*(1 + 5*x^2/(G(0) - 5*x^2)); G(k) = (2*k+2)*(2*k+3) + x^2*(4*k^2+8*k+5) - x^2*(2*k+2)*(2*k+3)*(4*k^2+16*k+17)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 19 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 + (4*k^2+4*k+2)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) ~ (-1)^(n+1) * cosh(Pi/2) * 2^(2*n-1) * n^(2*n-2) / exp(2*n). - Vaclav Kotesovec, Oct 23 2013
|a(n+2)| = Product_{k=0..n} ((2k+1)^2+1). - Andrew Slattery, Jul 03 2022

Name corrected by Andrew Slattery, Jul 03 2022

A166748 E.g.f.: exp(6*arcsin(x)).

Original entry on oeis.org

1, 6, 36, 222, 1440, 9990, 74880, 609390, 5391360, 51798150, 539136000, 6060383550, 73322496000, 951480217350, 13198049280000, 195053444556750, 3061947432960000, 50908949029311750, 894088650424320000

Offset: 0

Jaume Oliver Lafont, Oct 21 2009

nonn

exp(6*arcsin(1/2)) is Aleksandr Gelfond's constant exp(Pi).

G. C. Greubel, Table of n, a(n) for n = 0..445
A. R. Povolotsky et al., With regards to OEIS A166748, sci.math.symbolic usenet group, 2009

Cf. A166741, A006228, A039661.

Round[Table[3*2^(n-1)*(E^(3*Pi)-(-1)^n*E^(-3*Pi))*Abs[Gamma[n/2+3*I]]^2/Pi,{n,0,20}]] (* Vaclav Kotesovec, Nov 06 2014 *)
CoefficientList[Series[Exp[6*ArcSin[x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 06 2014 *)

A166748(n)=round(norm(gamma(n/2+3*I))/Pi*if(n%2,cosh(3*Pi),sinh(3*Pi))*3<M. F. Hasler, Oct 25 2009

PARI
```
a(n)=polcoeff(exp(6*asin(x)),n)*n!
```

a(n)=(1+5*(n%2))*prod(k=0,n\2-1,(2*k+n%2)^2+36) \\ Jaume Oliver Lafont, Oct 28 2009

Contribution from Alexander R. Povolotsky, Oct 24 2009: (Start)
a(n+2) = (n^2+36)*a(n), a(0)=1, a(1)=6.
The above recurrence leads to
a(n) = (3*2^n*gamma(-3*i+n/2)*gamma(3*i+n/2)*(cos((n*Pi)/2)+i*sin((n*Pi)/2))*sinh(((6-i*n)*Pi)/2))/Pi where "i" is imaginary unit. (End)
a(n) = 3*2^(n-1)*(exp(3*Pi)-(-1)^n*exp(-3*Pi))*|Gamma(n/2+3i)|^2/Pi. - R. J. Mathar and M. F. Hasler, Oct 25 2009
a(n) ~ 6 * (exp(3*Pi) - (-1)^n*exp(-3*Pi)) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 06 2014

Minor edits by Vaclav Kotesovec, Nov 06 2014

A091885 Triangle T(n,k) defined by the generating function cosh(sqrt(y)arcsin(x)) + sqrt(y)sinh(sqrt(y)arcsin(x)) - 1 = Sum_{n>=1} Sum_{k=1..n} T(n,k)y^k *x^n/n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 9, 10, 1, 64, 20, 1, 225, 259, 35, 1, 2304, 784, 56, 1, 11025, 12916, 1974, 84, 1, 147456, 52480, 4368, 120, 1, 893025, 1057221, 172810, 8778, 165, 1, 14745600, 5395456, 489280, 16368, 220, 1, 108056025, 128816766, 21967231, 1234948, 28743

Offset: 1

Karol A. Penson, Feb 08 2004

Row sums are equal to A006228(n). This is sequence A121408 without the intertwining zeros. - Emeric Deutsch, Jul 28 2006

This number triangle corresponds to the coefficients of the polynomial of the denominator of Fourier cosine coefficients for functions of the form sin(x)^(2*k) for integer n. For example (k=5), evaluating Integral_{x=-Pi..Pi} cos(n*x)*sin(x)^10 dx, we have -7257600*sin(n*Pi)/(-14745600*n + 5395456*n^3 - 489280*n^5 + 16368*n^7 - 220*n^9 + n^11); note the sequence of the coefficients of the polynomial of the denominator: -14745600, 5395456, -489280, 16368, -220, 1. - John M. Campbell, May 28 2011

			Triangle starts:
    1;
    1;
    1,   1;
    4,   1;
    9,  10,   1;
   64,  20,   1;
  225, 259,  35,   1;

Cf. A006228.

Cf. A121408.

G:=cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1: Gser:=simplify(series(G,x=0,15)): for n from 1 to 13 do P[n]:=sort(expand(n!*coeff(Gser,x,n))) od: for n from 1 to 13 do seq(coeff(P[n],y,k),k=1..ceil(n/2)) od; # yields sequence in triangular form # Emeric Deutsch, Jul 28 2006

m = 14; (* number of rows *)
T = Rest /@ Rest[CoefficientList[#, y]& /@ (CoefficientList[Cosh[Sqrt[y]* ArcSin[x]] + Sqrt[y]*Sinh[Sqrt[y]*ArcSin[x]] - 1  + O[x]^(m + 1), x]* Range[0, m]! // Simplify[#, y > 0]&)];
Flatten[T] (* Jean-François Alcover, Sep 27 2021 *)

E.g.f.: cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A166741 E.g.f.: exp(2*arcsin(x)).

Original entry on oeis.org

1, 2, 4, 10, 32, 130, 640, 3770, 25600, 199810, 1740800, 16983850, 181043200, 2122981250, 26794393600, 367275756250, 5358878720000, 84106148181250, 1393308467200000, 24643101417106250, 457005177241600000

Offset: 0

Jaume Oliver Lafont, Oct 21 2009

nonn

exp(2*arcsin(1)) is Aleksandr Gelfond's constant.

Wikipedia, Gelfond's constant

Cf. A006228, A039661, A166748.

seq(simplify(2^(n-1) * (cosh(Pi)*(1-(-1)^n) + sinh(Pi)*(1+(-1)^n)) * GAMMA((1/2)*n-I)*GAMMA((1/2)*n+I) / Pi), n=0..20); # Vaclav Kotesovec, Nov 06 2014

CoefficientList[Series[E^(2*ArcSin[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
FullSimplify[Table[2^(n-1) * (E^(Pi)-(-1)^n*E^(-Pi)) * Gamma[n/2-I] * Gamma[n/2+I] / Pi,{n,0,20}]] (* Vaclav Kotesovec, Nov 06 2014 *)

for (n=0,25,print(polcoeff(exp(2*asin(x)),n)*n!,","))

a(n) ~ 2 * n^(n-1) * (exp(Pi) - (-1)^n/exp(Pi)) / exp(n). - Vaclav Kotesovec, Aug 04 2014
From Vaclav Kotesovec, Nov 06 2014: (Start)
a(n) = (n^2 - 4*n + 8)*a(n-2).
a(n) = 2^(n-1) * (exp(Pi)-(-1)^n*exp(-Pi)) * GAMMA(n/2-I) * GAMMA(n/2+I) / Pi.
(End)

A259647 Expansion of exp(x*arcsin(x)) (even powers only).

Original entry on oeis.org

1, 2, 16, 294, 10424, 635130, 60535212, 8378845734, 1591416365520, 397329777218034, 126160335768212820, 49635257475383554590, 23694522124288261524984, 13490127107426613875639850, 9029074877857980800375629500, 7018229497764789751949369835030

Offset: 0

Vaclav Kotesovec, Jul 02 2015

nonn

Cf. A006228, A191301.

nmax=20; Table[(CoefficientList[Series[E^(x*ArcSin[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}]

a(n) ~ (2*n)! * exp(Pi/2) / (2*sqrt(Pi)*n^(3/2)).

cp's OEIS Frontend

A385343 Exponential Riordan array (1, arcsin(x)).

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A002019 a(n) = a(n-1) - (n-1)(n-2)a(n-2).

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Magma

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A101928 E.g.f. cos(arcsinh(x)) = sin(arccosh(x)) (even powers only).

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

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A385369 Expansion of e.g.f. x + sqrt(x^2 + 1).

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PARI

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A101927 E.g.f. of sin(arcsinh(x)) (odd powers only).

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Maple

Mathematica

Formula

Extensions

A166748 E.g.f.: exp(6*arcsin(x)).

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Mathematica

PARI

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!.

A091885 Triangle T(n,k) defined by the generating function cosh(sqrt(y)arcsin(x)) + sqrt(y)sinh(sqrt(y)arcsin(x)) - 1 = Sum_{n>=1} Sum_{k=1..n} T(n,k)y^k *x^n/n!.