A147309 -id:A147309 - cp's OEIS frontend

A012123 E.g.f.: exp(arcsin(tanh(x))).

1, 1, 1, 0, -3, -4, 21, 80, -263, -2224, 4841, 88960, -99723, -4942144, -199939, 366928640, 501445617, -35219691264, -101818966319, 4251941253120, 19731909099757, -631113275843584, -4192563651606299, 113005305852006400, 1009030667701246697

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

sign

			exp(arcsin(tanh(x))) = 1 + x + 1/2!*x^2 - 3/4!*x^4 - 4/5!*x^5 + 21/6!*x^6 ...

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, i]*EulerE[i]*a[n-i-1], {i, 0, n-1}]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, May 22 2017, after Vladimir Kruchinin *)

a(n):=sum(sum((sum((-1)^((3*k)/2)*(sum((2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!,i,0,k))*binomial((r-2)/2,(r-m-k)/2),k,0,r-m))*((-1)^(r-m)+1)*sum(binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k),k,r,n),r,m,n),m,1,n)/2; /* Vladimir Kruchinin, Jun 09 2011 */

a(n):=if n=0 then 1 else sum(binomial(n-1,i)*euler(i)*a(n-i-1),i,0,n-1); /* Vladimir Kruchinin, Feb 26 2015 */

x='x+O('x^66); Vec(serlaplace(exp(asin(tanh(x))))) \\ Joerg Arndt, Feb 26 2015

a(n) = (-i)^n * Z(n,i), where i = sqrt(-1) and Z(n,x) denotes the n-th zigzag polynomial as described in A147309. Alternative form of the egf: {sec(i*x) - tan(i*x)}^i. - Peter Bala, Jan 26 2011
a(n)=sum(m=1..n, sum(r=m..n, (sum(k=r..n, (-1)^((3*k)/2)*(sum(i=0..k, (2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!))*binomial((r-2)/2,(r-m-k)/2)))*((-1)^(r-m)+1)*sum(k=0..r-m, binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k))))/2, n>0, a(0)=1. - Vladimir Kruchinin, Jun 09 2011
a(n) = sum(i=0..n-1, binomial(n-1,i)*euler(i)*a(n-i-1)), a(0)=1. - Vladimir Kruchinin, Feb 26 2015

A147308 Riordan array [sech(x), arcsin(tanh(x))].

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -4, 0, 1, 5, 0, -10, 0, 1, 0, 40, 0, -20, 0, 1, -61, 0, 175, 0, -35, 0, 1, 0, -768, 0, 560, 0, -56, 0, 1, 1385, 0, -4996, 0, 1470, 0, -84, 0, 1, 0, 24320, 0, -22720, 0, 3360, 0, -120, 0, 1, -50521, 0, 214445, 0

Offset: 0

Paul Barry, Nov 05 2008

Production array is [cos(x),x] beheaded. Inverse is A147309. Row sums are A012123(n+1).
If signs are ignored this is identical to A147309. - N. J. A. Sloane, Nov 07 2008
The Bell transform of the Euler numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins:
    1;
    0,   1;
   -1,   0,   1;
    0,  -4,   0,   1;
    5,   0, -10,   0,   1;
    0,  40,   0, -20,   0,   1;
  -61,   0, 175,   0, -35,   0,   1;

Nathaniel Johnston, Table of n, a(n) for n = 0..229

Z := proc(n,x) option remember; if n = 0 then return 1: else return 1/2*x*(Z(n-1, x-1)+Z(n-1, x+1)): fi:end:
with(PolynomialTools): for n from 1 to 10 do for k from 1 to n do printf("%d, ",(-1)^floor((n-k)/2)*coeff(Z(n,x), x, k)):od: printf("\n"):od: # Nathaniel Johnston, Apr 21 2011

t[n_, k_] := SeriesCoefficient[ 2^k*ArcTan[(E^x - 1)/(E^x + 1)]^k*n!/k!, {x, 0, n}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 23 2015 *)
Z[n_, x_] := Z[n, x] = If[n == 0, 1, x*(Z[n-1, x-1] + Z[n-1, x+1])/2 // Simplify]; t[n_, k_] := (-1)^Floor[(n-k)/2]*Coefficient[Z[n, x], x, k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 27 2015, after Maple *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(euler_number, 10) # Peter Luschny, Jan 18 2016

From Vladimir Kruchinin, Dec 18 2011: (Start)
[gd(x)]^m = sum(n>=m T(n,m)*m!/n!*x^n), where gd(x) is the Gudermannian function.
T(n,m) = sum(j=0..(n-m)/2, (sum(i=0..2*j, (2^(i)*Stirling1(i+m,m) *C(2*j+m-1,i+m-1))/(i+m)!)) *sum(k=0..n-2*j-m, (-1)^(k+j) *C(k+2*j+m-1,2*j+m-1) *(k+2*j+m)! *2^(-k-2*j) *Stirling2(n,k+2*j+m))), n>=m>=1.
(End)

A147312 Riordan array [1,log(sec(x)+tan(x))].

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 5, 0, 10, 0, 1, 0, 0, 40, 0, 20, 0, 1, 0, 61, 0, 175, 0, 35, 0, 1, 0, 0, 768, 0, 560, 0, 56, 0, 1, 0, 1385, 0, 4996, 0, 1470, 0, 84, 0, 1, 0, 0, 24320, 0, 22720, 0, 3360, 0, 120, 0, 1, 0, 50521, 0, 214445, 0, 81730, 0, 6930, 0, 165, 0, 1

Offset: 0

Paul Barry, Nov 05 2008

Row sums are A000111. Inverse is A147311.
Production array is [cosh(x),x] with a column of 0's prepended.
The product [sec(x),x]*A147312 is A147309.
Apart from signs, same as A147311. - N. J. A. Sloane, Nov 07 2008
Also the Bell transform of the absolute Euler numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 0, 1,
0, 0, 4, 0, 1,
0, 5, 0, 10, 0, 1,
0, 0, 40, 0, 20, 0, 1

# The function BellMatrix is defined in A264428.
BellMatrix(n -> abs(euler(n)), 10); # Peter Luschny, Jan 29 2016

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[Abs[EulerE[#]] &, rows];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

T(n,m)=sum(k=m..n, A147315(n,k)*stirling1(k,m)), n>0,k>0, T(0,0)=1, T(0,k)=0, k>0. [From Vladimir Kruchinin, Mar 10 2011]

More terms from Jean-François Alcover, Jun 28 2018

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A012123 E.g.f.: exp(arcsin(tanh(x))).

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A147308 Riordan array [sech(x), arcsin(tanh(x))].

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A147312 Riordan array [1,log(sec(x)+tan(x))].

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