A060524 -id:A060524 - cp's OEIS frontend

A111594 Triangle of arctanh numbers.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 24, 0, 20, 0, 1, 0, 0, 184, 0, 40, 0, 1, 0, 720, 0, 784, 0, 70, 0, 1, 0, 0, 8448, 0, 2464, 0, 112, 0, 1, 0, 40320, 0, 52352, 0, 6384, 0, 168, 0, 1, 0, 0, 648576, 0, 229760, 0, 14448, 0, 240, 0, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Sheffer triangle associated to Sheffer triangle A060524.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
The inverse matrix of A with elements a(n,m), n,m>=0, is given in A111593.
In the umbral calculus notation (see the S. Roman reference) this triangle would be called associated to (1,tanh(y)).
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060524 satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
Without the n=0 row and m=0 column and signed, this will become the Jabotinsky triangle A049218 (arctan numbers). For Jabotinsky matrices see the Knuth reference under A039692.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*arctanh(y)).
Exponential Riordan array [1, arctanh(x)] = [1, log(sqrt((1+x)/(1-x)))]. - Paul Barry, Apr 17 2008
Also the Bell transform of A005359. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Binomial convolution of row polynomials:
p(3,x)= 2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1,
together with those from A060524:
s(3,x)= 5*x+x^3; s(2,x)= 1+x^2, s(1,x)= x, s(0,x)= 1; therefore:
5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = 2*y+y^3 + 3*x*y^2 + 3*(1+x^2)*y + (5*x+x^3).
Triangle begins:
  1;
  0,   1;
  0,   0,    1;
  0,   2,    0,   1;
  0,   0,    8,   0,    1;
  0,  24,    0,  20,    0,  1;
  0,   0,  184,   0,   40,  0,   1;
  0, 720,    0, 784,    0, 70,   0, 1;
  0,   0, 8448,   0, 2464,  0, 112, 0, 1;
...

Wolfdieter Lang, First 10 rows.

Row sums: A000246.

Cf. A005359, A049218, A060524, A111593.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::even, n!, 0), 10); # Peter Luschny, Jan 27 2016

rows = 10;
t = Table[If[EvenQ[n], n!, 0], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

# uses[riordan_array from A256893]
riordan_array(1, atanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015

E.g.f. for column m>=0: ((arctanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((arctanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) + (n-2)*(n-1)*a(n-2, m), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n

A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -8, 0, 1, 24, 0, -20, 0, 1, 0, 184, 0, -40, 0, 1, -720, 0, 784, 0, -70, 0, 1, 0, -8448, 0, 2464, 0, -112, 0, 1, 40320, 0, -52352, 0, 6384, 0, -168, 0, 1, 0, 648576, 0, -229760, 0, 14448, 0, -240, 0, 1, -3628800, 0, 5360256, 0, -804320, 0, 29568, 0, -330, 0, 1

Offset: 1

N. J. A. Sloane

|T(n,k)| gives the sum of the M_2 multinomial numbers (A036039) for those partitions of n with exactly k odd parts. E.g.: |T(6,2)| = 144 + 40 = 184 from the partitions of 6 with exactly two odd parts, namely (1,5) and (3,3), with M_2 numbers 144 and 40. Proof via the general Jabotinsky triangle formula for |T(n,k)| using partitions of n into k parts and their M_3 numbers (A036040). Then with the special e.g.f. of the (unsigned) k=1 column, f(x):= arctanh(x), only odd parts survive and the M_3 numbers are changed into the M_2 numbers. For the Knuth reference on Jabotinsky triangles see A039692. - Wolfdieter Lang, Feb 24 2005 [The first two sentences have been corrected thanks to the comment by José H. Nieto S. given below. - Wolfdieter Lang, Jan 16 2012]
|T(n,k)| gives the number of permutations of {1,2,...,n} (degree n permutations) with the number of odd cycles equal to k. E.g.: |T(5,3)|= 20 from the 20 degree 5 permutations with cycle structure (.)(.)(...). Proof: Use the cycle index polynomial for the symmetric group S_n (see the M_2 array A036039 or A102189) together with the partition interpretation of |T(n,k)| given above. - Wolfdieter Lang, Feb 24 2005 [See the following José H. Nieto S. correction. - Wolfdieter Lang, Jan 16 2012]
The first sentence of the above comment is inexact, it should be "|T(n,k)| gives the number of degree n permutations which decompose into exactly k odd cycles". The number of degree n permutations with k odd cycles (and, possibly, other cycles of even length) is given by A060524. - José H. Nieto S., Jan 15 2012
The unsigned triangle with e.g.f. exp(x*arctanh(z)) is the associated Jabotinsky type triangle for the Sheffer type triangle A060524. See the comments there. - Wolfdieter Lang, Feb 24 2005
Also the Bell transform of the sequence (-1)^(n/2)*A005359(n) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
   1;
   0,   1;
  -2,   0,   1;
   0,  -8,   0,   1;
  24,   0, -20,   0,   1;
   0, 184,   0, -40,   0,   1;
  ...
O.g.f. for fifth subdiagonal: (24*t+16*t^2)/(1-t)^7 = 24*t + 184*t^2 + 784*t^3 + 2404*t^4 + ....

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial

Essentially same as A008309, which is the main entry for this sequence.

Row sums (unsigned) give A000246(n); signed row sums give A002019(n), n>=1. A137513.

A049218 := proc(n,k)(-1)^((3*n+k)/2) *add(2^(j-k)*n!/j! *stirling1(j,k) *binomial(n-1,j-1),j=k..n) ; end proc: # R. J. Mathar, Feb 14 2011
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::odd, 0, (-1)^(n/2)*n!), 10); # Peter Luschny, Jan 28 2016

t[n_, k_] := (-1)^((3n+k)/2)*Sum[ 2^(j-k)*n!/j!*StirlingS1[j, k]*Binomial[n-1, j-1], {j, k, n}]; Flatten[ Table[ t[n, k], {n, 1, 11}, {k, 1, n}]] (* Jean-François Alcover, Dec 06 2011, after Vladimir Kruchinin *)
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 23 2018, after Peter Luschny *)

T(n,k)=polcoeff(serlaplace(atan(x)^k/k!), n)

E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(n, k) x^n/n!.
T(n,k) = ((-1)^((3*n+k)/2)*n!/2^k)*Sum_{i=k..n} 2^i*binomial(n-1,i-1)*Stirling1(i,k)/i!. - Vladimir Kruchinin, Feb 11 2011
E.g.f.: exp(t*arctan(x)) = 1 + t*x + t^2*x^2/2! + t*(t^2-2)*x^3/3! + .... The unsigned row polynomials are the Mittag-Leffler polynomials M(n,t/2). See A137513. The compositional inverse (with respect to x) (x-t/2*log((1+x)/(1-x)))^(-1) = x/(1-t) + 2*t/(1-t)^4*x^3/3!+ (24*t+16*t^2)/(1-t)^7*x^5/5! + .... The rational functions in t generate the (unsigned) diagonals of the table. See the Bala link. - Peter Bala, Dec 04 2011

Additional comments from Michael Somos

A060081 Exponential Riordan array (sech(x), tanh(x)).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -5, 0, 1, 5, 0, -14, 0, 1, 0, 61, 0, -30, 0, 1, -61, 0, 331, 0, -55, 0, 1, 0, -1385, 0, 1211, 0, -91, 0, 1, 1385, 0, -12284, 0, 3486, 0, -140, 0, 1, 0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1, -50521, 0, 663061

Offset: 0

Wolfdieter Lang, Mar 29 2001

Previous name was: "Triangle of coefficients (lower triangular matrix) of certain (binomial) convolution polynomials related to 1/cosh(x) and tanh(x). Use trigonometric functions for the unsigned version".
Row sums give A009265(n) (signed); A009244(n) (unsigned). Column sequences without interspersed zeros and unsigned: A000364 (Euler), A000364, A060075-8 for m=0,...,5.
a(n,m) = ((-1)^((n-m)/2))*ay(m+1,(n-m)/2) if n-m is even, else 0; where the rectangular array ay(n,m) is defined in A060058 Formula.
The row polynomials p(n,x) appear in a problem of thermo field dynamics (Bogoliubov transformation for the harmonic Bose oscillator). See the link to a .ps.gz file where they are called R_{n}(x).
The inverse of this Sheffer matrix with elements a(n,m) is the Sheffer matrix A060524. This Sheffer triangle appears in the Moyal star product of the harmonic Bose oscillator: x^{*n} = Sum_{m=0..n} a(n,m) x^m with x = 2 (bar a) a/hbar. See the Th. Spernat link, pp. 28, 29, where the unsigned version is used for y=-ix. - Wolfdieter Lang, Jul 22 2005
In the umbral calculus (see Roman reference under A048854) the p(n,x) are called Sheffer for (g(t)=1/cosh(arctanh(t)) = 1/sqrt(1-t^2), f(t)=arctanh(t)).
p(n,x) := Sum_{m=0..n} a(n,m)*x^m, n >= 0, are monic polynomials satisfying p(n,x+y) = Sum_{k=0..n} binomial(n,k)*p(k,x)*q(n-k,y) (binomial, also called exponential, convolution polynomials) with the row polynomials of the associated triangle q(n,x) := Sum_{m=0..n} A111593(n,m)*x^m. E.g.f. for p(n,x) is exp(x*tanh(z))*cosh(z)(signed). [Corrected by Wolfdieter Lang, Sep 12 2005]
Exponential Riordan array [sech(x), tanh(x)]. Unsigned triangle is [sec(x), tan(x)]. - Paul Barry, Jan 10 2011

			p(3,x) = -5*x + x^3.
Exponential convolution together with A111593 for row polynomials q(n,x), case n=2: -1+(x+y)^2 = p(2,x+y) = 1*p(0,x)*q(2,y) + 2*p(1,x)*q(1,y) + 1*p(2,x)*q(0,y) = 1*1*y^2 + 2*x*y + 1*(-1+x^2)*1.
Triangle begins:
  1,
  0, 1,
  -1, 0, 1,
  0, -5, 0, 1,
  5, 0, -14, 0, 1,
  0, 61, 0, -30, 0, 1,
  -61, 0, 331, 0, -55, 0, 1,
  0, -1385, 0, 1211, 0, -91, 0, 1,
  1385, 0, -12284, 0, 3486, 0, -140, 0, 1,
  0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1,
  -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1,
  ...
As a right-aligned triangle:
                                                       1;
                                                    0, 1;
                                                -1, 0, 1;
                                           0,   -5, 0, 1;
                                        5, 0,  -14, 0, 1;
                                 0,    61, 0,  -30, 0, 1;
                            -61, 0,   331, 0,  -55, 0, 1;
                     0,   -1385, 0,  1211, 0,  -91, 0, 1;
               1385, 0,  -12284, 0,  3486, 0, -140, 0, 1;
          0,  50521, 0,  -68060, 0,  8526, 0, -204, 0, 1;
  -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1;
  ...
Production matrix begins
   0,   1;
  -1,   0,   1;
   0,  -4,   0,   1;
   0,   0,  -9,   0,   1;
   0,   0,   0, -16,   0,   1;
   0,   0,   0,   0, -25,   0,   1;
   0,   0,   0,   0,   0, -36,   0,   1;
   0,   0,   0,   0,   0,   0, -49,   0,   1;
   0,   0,   0,   0,   0,   0,   0, -64,   0,   1;
- _Paul Barry_, Jan 10 2011

W. Lang, Two normal ordering problems and certain Sheffer polynomials, in Difference Equations, Special Functions and Orthogonal Polynomials, edts. S. Elaydi et al., World Scientific, 2007, pages 354-368. [From Wolfdieter Lang, Feb 06 2009]

Paul Barry, Exponential Riordan arrays and permutation enumeration,Journal of Integer Sequences, Vol. 13 (2010)
Wolfdieter Lang, Thermo field dynamics, exercise 29. WS 2008/2009 (in German)
Th. Spernat, Diplomarbeit 2004 (in German) (with permission)

riordan := (d,h,n,k) -> coeftayl(d*h^k,x=0,n)*n!/k!:
A060081 := (n,k) -> riordan(sech(x),tanh(x),n,k):
seq(print(seq(A060081(n,k),k=0..n)),n=0..5); # Peter Luschny, Apr 15 2015

max = 12; t = Transpose[ Table[ PadRight[ CoefficientList[ Series[ Tanh[x]^m/m!/Cosh[x], {x, 0, max}], x], max + 1, 0]*Table[k!, {k, 0, max}], {m, 0, max}]]; Flatten[ Table[t[[n, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Sep 29 2011 *)

def A060081_triangle(dim): # computes unsigned T(n, k).
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+1)^2*M[n-1,k+1]
    return M
A060081_triangle(9) # Peter Luschny, Sep 19 2012

E.g.f. for column m: (((tanh(x))^m)/m!)/cosh(x), m >= 0. Use trigonometric functions for unsigned case.
a(n, m) = a(n-1, m-1)-((m+1)^2)*a(n-1, m+1); a(0, 0)=1; a(n, -1) := 0, a(n, m)=0 if n < m. Use sum of the two recursion terms for unsigned case.
a(n, k) = (1/(k+1)!)*Sum_{q=0..n} C(n,q)*((-1)^(n-q)+1)*((-1)^(q-k)+1)*Sum_{j=0..q-k} C(j+k,k)*(j+k+1)!*2^(q-j-k-2)*(-1)^j*Stirling2(q+1,j+k+1). - Vladimir Kruchinin, Feb 12 2019

New name (using a comment from Paul Barry) from Peter Luschny, Apr 15 2015

A060338 Triangle T(n,k) of coefficients of Meixner polynomials of degree n, k=0..n.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 5, 0, 1, 0, 14, 0, 9, 1, 0, 30, 0, 89, 0, 1, 0, 55, 0, 439, 0, 225, 1, 0, 91, 0, 1519, 0, 3429, 0, 1, 0, 140, 0, 4214, 0, 24940, 0, 11025, 1, 0, 204, 0, 10038, 0, 122156, 0, 230481, 0, 1, 0, 285, 0, 21378, 0, 463490, 0, 2250621, 0, 893025

Offset: 0

Vladeta Jovovic, Mar 30 2001

The Meixner polynomials M_n(x) satisfy the recurrence: M_(k+1)=x*M_k-k^2*M_(k-1), M_(-1)=0, M_0=1.
See A060524 for an application to combinatorics. - N. J. A. Sloane, May 30 2013
The Meixner polynomials M_n(x) satisfy: M_n(x)=n!*sum(m=0..n/2, binomial(2*m,m)*sum(j=m..n/2, (-1)^(j)*x^(n-2*j)*sum(i=0..2*j-2*m, (2^(i-2*m)*stirling1(i+n+(-2)*j,n-2*j)*binomial(n-2*m-1,2*j-2*m-i))/(i+n+(-2)*j)!))). [Vladimir Kruchinin, Sep 25 2013]

			[1],
[1, 0],
[1, 0, -1],
[1, 0, -5, 0],
[1, 0, -14, 0, 9],
[1, 0, -30, 0, 89, 0],
[1, 0, -55, 0, 439, 0, -225],
[1, 0, -91, 0, 1519, 0, -3429, 0],
[1, 0, -140, 0, 4214, 0, -24940, 0, 11025],
[1, 0, -204, 0, 10038, 0, -122156, 0, 230481, 0], ...
M_1(x)=x, M_2(x)=x^2-1, M_3(x)=x^3-5*x, M_4(x)=x^4-14*x^2+9, M_5(x)=x^5-30*x^3+89*x, M_6(x)=x^6-55*x^4+439*x^2-225,...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
A. Hamdi and J. Zeng, Orthogonal polynomials and operator orderings, J. Math. Phys., 51:043506, 2010; arXiv:1006.0808 [math.CO], 2010.
R. J. Mathar, Gaussian Quadrature of the Integrals Int_(-infty)^infty F(x) dx /cosh(x)
J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.

Cf. A028353, A060524, A000330 (third column), A214615 (row sums), A214616 (fifth column).

Triangle without zeros: A094368. Unsigned version: A060524.

m[0] = 1; m[1] = x; m[k_] := m[k] = x*m[k - 1] - (k - 1)^2*m[k - 2]; row[n_] := CoefficientList[m[n], x] // Reverse // Abs; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)

M(n,x):=n!*sum(binomial(2*m,m)*sum(((sum((2^(i-2*m)*stirling1(i+n-2*j,n-2*j)*binomial(n-2*m-1,2*j-2*m-i))/(i+n-2*j)!,i,0,2*j-2*m))*(-1)^(j)*x^(n-2*j)),j,m,n/2),m,0,n/2); /* Vladimir Kruchinin, Sep 25 2013 */

E.g.f.: exp(x*arctan(y))/sqrt(1+y^2).

A028353 Coefficient of x^(2n+1) in arctanh(x)/sqrt(1-x^2), multiplied by (2n+1)!.

Original entry on oeis.org

1, 5, 89, 3429, 230481, 23941125, 3555578025, 715154761125, 187188449198625, 61836509511685125, 25163273966324405625, 12368068140988819153125, 7224011282550809645600625

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Number of degree-(2*n+1) permutations with exactly one odd cycle. - Vladeta Jovovic, Aug 13 2004

a(n)=sum over all multinomials M2(2*n+1,k), k from {1..p(2*n+1)} restricted to partitions with exactly one odd and possibly even parts. p(2*n+1)= A000041(2*n+1) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n+1,k). - Wolfdieter Lang, Aug 07 2007.

			arctanh(x)/sqrt(1-x^2) = x + 5/6*x^3 + 89/120*x^5 + 381/560*x^7 + ...
Multinomial representation for a(2): partitions of 2*2+1=5 with one odd part: (5) with position k=1, (1,4) with k=2, (2,3) with k=3, (1,2^2) with k=5; M2(5,1)= 24, M2(5,2)= 30, M2(5,3)= 20, M2(5,5)= 15, adding up to a(2)=89.

G. C. Greubel, Table of n, a(n) for n = 0..220
Zhi-Hong Sun, Congruences for the Apéry numbers modulo p^3, arXiv:2409.06544 [math.NT], 2024. See t(n).

Cf. A060338.

Cf. A060524.

Table[n!*SeriesCoefficient[ArcTanh[x]/Sqrt[1-x^2],{x,0,n}],{n,1,41,2}] (* Vaclav Kotesovec, Oct 24 2013 *)

D-finite with recurrence: a(n) = (8*n^2 - 4*n + 1)*a(n-1) - 4*(n-1)^2*(2*n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 24 2013

a(n) ~ (2*n)^(2*n+1)*log(n)/exp(2*n) * (1 + (gamma + 4*log(2)) / log(n)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 24 2013

A060523 Triangle T(n,k) = number of degree-n permutations with k even cycles, k=0..n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 3, 3, 0, 0, 9, 12, 3, 0, 0, 45, 60, 15, 0, 0, 0, 225, 345, 135, 15, 0, 0, 0, 1575, 2415, 945, 105, 0, 0, 0, 0, 11025, 18480, 9030, 1680, 105, 0, 0, 0, 0, 99225, 166320, 81270, 15120, 945, 0, 0, 0, 0, 0, 893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0

Offset: 0

Vladeta Jovovic, Apr 01 2001

			Triangle T(n,k) begins:
       1;
       1,       0;
       1,       1,      0;
       3,       3,      0,      0;
       9,      12,      3,      0,     0;
      45,      60,     15,      0,     0,   0;
     225,     345,    135,     15,     0,   0, 0;
    1575,    2415,    945,    105,     0,   0, 0, 0;
   11025,   18480,   9030,   1680,   105,   0, 0, 0, 0;
   99225,  166320,  81270,  15120,   945,   0, 0, 0, 0, 0;
  893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0;
  ...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 189, Exercise 3.3.13.

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A000246, A096471.
Row sums give A000142.
Cf. A060524, A092691.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
      `if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015

nn = 6; Range[0, nn]! CoefficientList[
   Series[(1 - x^2)^(-y/2) ((1 + x)/(1 - x))^(1/2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 27 2012 *)

E.g.f.: (1+x)^((1-y)/2)/(1-x)^((1+y)/2).

Sum_{k=0..n} k * T(n,k) = A092691(n). - Alois P. Heinz, Aug 17 2023

Showing 1-10 of 13 results. Next

a(n) = sum over all multinomials M2(2*(n+1),k), k from {1..p(2*(n+1))} restricted to partitions with exactly two odd and any nonnegative number even parts. p(2*(n+1)) = A000041(2*(n+1)) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*(n+1),k). - Wolfdieter Lang, Aug 07 2007

a(n) = sum over all multinomials M2(2*n+3,k), k from {1..p(2*n+3)} restricted to partitions with exactly three odd and any nonnegative number of even parts. p(2*n+3)= A000041(2*n+3) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n,k). - Wolfdieter Lang, Aug 07 2007

a(n) = sum over all M2(2*n+4,k), k from {1..p(2*n+4)} restricted to partitions with exactly four odd and any nonnegative number of even parts. p(2*n+4)= A000041(2*n+4) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n+4,k). - Wolfdieter Lang, Aug 07 2007

a(n) = sum over all M2(2*n+5,k), k from {1..p(2*n+5)} restricted to partitions with exactly five odd and possibly even parts. p(2*n+5) = A000041(2*n+5) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n+5,k).

cp's OEIS Frontend

A103916 Column k=2 sequence (without zero entries) of table A060524.

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A103917 Column k=3 sequence (without zero entries) of table A060524.

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A103918 Column k=4 sequence (without zero entries) of table A060524.

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A131442 Sixth column (m=5) of triangle A060524 without zeros.

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A111594 Triangle of arctanh numbers.

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A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

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A060081 Exponential Riordan array (sech(x), tanh(x)).

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A060338 Triangle T(n,k) of coefficients of Meixner polynomials of degree n, k=0..n.

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A028353 Coefficient of x^(2n+1) in arctanh(x)/sqrt(1-x^2), multiplied by (2n+1)!.