A035342 -id:A035342 - cp's OEIS frontend

A035101 E.g.f. x(c(x/2)-1)/(1-2x), where c(x) = g.f. for Catalan numbers A000108.

0, 1, 9, 87, 975, 12645, 187425, 3133935, 58437855, 1203216525, 27125492625, 664761133575, 17600023616175, 500706514833525, 15234653491682625, 493699195087473375, 16977671416936605375, 617528830880480644125, 23687738668934964248625

Offset: 1

Wolfdieter Lang

2nd column of triangular array A035342 whose first column is given by A001147(n), n >= 1. Recursion: a(n) = 2*n*a(n-1)+ A001147(n-1), n >= 2, a(1)=0.
a(n) gives the number of organically labeled forests (sets) with two rooted ordered trees with n non-root vertices. See the example a(3)=9 given in A035342. Organic labeling means that the vertex labels along the (unique) path from the root to any of the leaves (degree 1, non-root vertices) is increasing. - Wolfdieter Lang, Aug 07 2007
a(n), n>=2, enumerates unordered n-vertex forests composed of two plane (ordered) ternary (3-ary) trees with increasing vertex labeling. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly 1 of the remaining n-1 chords are contained within the marked chord, see [Young]. - Donovan Young, Aug 11 2020

			a(2)=1 for the forest: {r1-1, r2-2} (with root labels r1 and r2). The order between the components of the forest is irrelevant (like for sets).
a(3)=9 increasing ternary 2-forest with n=3 vertices: there are three 2-forests (the one vertex tree together with any of the three different 2-vertex trees) each with three increasing labelings. - _Wolfdieter Lang_, Sep 14 2007

Robert Israel, Table of n, a(n) for n = 1..370
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A000108, A002866, A008549, A336599.

Cf. A001147 (m=1 column of A035342). See a D. Callan comment there on the number of increasing ordered rooted trees on n+1 vertices.

I:=[0,1,9]; [n le 3 select I[n] else - 2*(n-1)*(2*n-3)*Self(n-2)+(4*n-3)*Self(n-1): n in [1..30]]; // Vincenzo Librandi, Sep 12 2015

F:= gfun:-rectoproc({(4*n^2+6*n+2)*a(n)+(-4*n-5)*a(n+1)+a(n+2),a(1)=0,a(2)=1,a(3)=9},a(n),remember):
map(f, [$1..30]); # Robert Israel, Sep 11 2015

Table[Round [n! (4^(n - 1) - Binomial[2 n, n]/2)/2^(n - 1)], {n, 1, 20}] (* Vincenzo Librandi, Sep 12 2015 *)

a(n) = n!*(4^(n-1)-binomial(2*n, n)/2)/2^(n-1);
vector(40, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = n!*A008549(n-1)/2^(n-1) = n!(4^(n-1)-binomial(2*n, n)/2)/2^(n-1).
a(n) = (2n-2)*a(n-1) + A129890(n-2). - Philippe Deléham, Oct 28 2013
a(n) = n!*2^(n-1) - A001147(n) = A002866(n) - A001147(n). - Peter Bala, Sep 11 2015
a(n) = -2*(n-1)*(2*n-3)*a(n-2)+(4*n-3)*a(n-1). - Robert Israel, Sep 11 2015

A143173 Partition number array, called M32(-3), related to A000369(n,m) = |S2(-3;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 21, 9, 1, 231, 84, 27, 18, 1, 3465, 1155, 630, 210, 135, 30, 1, 65835, 20790, 10395, 4410, 3465, 3780, 405, 420, 405, 45, 1, 1514205, 460845, 218295, 169785, 72765, 72765, 30870, 19845, 8085, 13230, 2835, 735, 945, 63, 1, 40883535, 12113640, 5530140, 4074840

Offset: 1

Wolfdieter Lang, Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-3;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+2)-ary trees if the outdegree is r >= 0.
If M32(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle A000369(n,m) = |S2(-3;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

			a(4,3)=27. The relevant partition of 4 is (2^2). The 12 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are ternary because r=1 vertices are ternary (3-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4 versions due to the two ternary root vertices.

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A143172 (M32(-2) array), A144267 (M32(-4) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-3,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-3,j,1)|^e(n,k,j),j=1..n), with |S2(-3,n,1)|= A008545(n-1) = (4*n-5)(!^4) (4-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

Original entry on oeis.org

1, 1, 1, 3, 6, 1, 15, 45, 15, 1, 105, 420, 210, 28, 1, 945, 4725, 3150, 630, 45, 1, 10395, 62370, 51975, 13860, 1485, 66, 1, 135135, 945945, 945945, 315315, 45045, 3003, 91, 1, 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1, 34459425

Offset: 0

Paul Barry, Apr 12 2010

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
Diagonal sums are essentially A025164.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
(End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016

			Triangle begins
        1,
        1,        1,
        3,        6,        1,
       15,       45,       15,       1,
      105,      420,      210,      28,       1,
      945,     4725,     3150,     630,      45,      1,
    10395,    62370,    51975,   13860,    1485,     66,    1,
   135135,   945945,   945945,  315315,   45045,   3003,   91,   1,
  2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
Production matrix is
  1,  1,
  2,  5,  1,
  0, 12,  9,  1,
  0,  0, 30, 13,  1,
  0,  0,  0, 56, 17,   1,
  0,  0,  0,  0, 90,  21,   1,
  0,  0,  0,  0,  0, 132,  25,   1,
  0,  0,  0,  0,  0,   0, 182,  29,  1,
  0,  0,  0,  0,  0,   0,   0, 240, 33, 1.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
Tom Copeland, Juggling Zeros in the Matrix (Example II), 2020.

Cf. A113278.

Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.

ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)

Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
T(n,k) = A122848(2n,k+n). - R. J. Mathar, Jan 14 2011
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012
E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013
The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014
The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015
T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 2020

A370419 A(n, k) = 2^n*Pochhammer(k/2, n). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 8, 3, 1, 0, 105, 48, 15, 4, 1, 0, 945, 384, 105, 24, 5, 1, 0, 10395, 3840, 945, 192, 35, 6, 1, 0, 135135, 46080, 10395, 1920, 315, 48, 7, 1, 0, 2027025, 645120, 135135, 23040, 3465, 480, 63, 8, 1

Offset: 0

Peter Luschny, Mar 04 2024

			The array starts:
[0] 1,   1,    1,     1,     1,     1,     1,      1,      1, ...
[1] 0,   1,    2,     3,     4,     5,     6,      7,      8, ...
[2] 0,   3,    8,    15,    24,    35,    48,     63,     80, ...
[3] 0,  15,   48,   105,   192,   315,   480,    693,    960, ...
[4] 0, 105,  384,   945,  1920,  3465,  5760,   9009,  13440, ...
[5] 0, 945, 3840, 10395, 23040, 45045, 80640, 135135, 215040, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   3,   2,   1;
[4] 0,  15,   8,   3,  1;
[5] 0, 105,  48,  15,  4, 1;
[6] 0, 945, 384, 105, 24, 5, 1;
.
From _Werner Schulte_, Mar 07 2024: (Start)
Illustrating the LU decomposition of A:
    / 1                \   / 1 1 1 1 1 ... \   / 1   1   1   1    1 ... \
    | 0   1            |   |   1 2 3 4 ... |   | 0   1   2   3    4 ... |
    | 0   3   2        | * |     1 3 6 ... | = | 0   3   8  15   24 ... |
    | 0  15  18   6    |   |       1 4 ... |   | 0  15  48 105  192 ... |
    | 0 105 174 108 24 |   |         1 ... |   | 0 105 384 945 1920 ... |
    | . . .            |   | . . .         |   | . . .                  |. (End)

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Columns: A000007, A001147, A000165, A001147 (shifted), A002866, A051577, A051578, A051579, A051580.
Rows: A000012, A001477, A005563, A370912, A190577.
Cf. A035342, A265649, A370890, A370982 (row sums of the triangle), A370915, A371025, A371077.

A := (n, k) -> 2^n*pochhammer(k/2, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 2*x)^(-k/2);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..8);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-2)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the comment of Werner Schulte about the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371025(n - 1,  k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));  #  Peter Luschny, Mar 08 2024

A370419[n_, k_] := 2^n*Pochhammer[k/2, n];
Table[A370419[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 06 2024 *)

def A(n, k): return 2**n * rising_factorial(k/2, n)
for n in range(6): print([A(n, k) for k in range(9)])

The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-2)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
From Werner Schulte, Mar 07 2024: (Start)
A(n, k) = Product_{i=1..n} (2*i - 2 + k).
E.g.f. of column k: Sum_{n>=0} A(n, k) * t^n / (n!) = (1/sqrt(1 - 2*t))^k.
A(n, k) = A(n+1, k-2) / (k - 2) for k > 2.
A(n, k) = Sum_{i=0..k-1} i! * A265649(n, i) * binomial(k-1, i) for k > 0.
E.g.f. of row n > 0: Sum_{k>=1} A(n, k) * x^k / (k!) = (Sum_{k=1..n} A035342(n, k) * x^k) * exp(x).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (k! * n!) = exp(x/sqrt(1 - 2*t)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1 / (1 - x/sqrt(1 - 2*t)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(n, k) = A035342(n, k) * k! for 1 <= k <= n and L(n, 0) = 0^n. Note that L(n, k) + L(n, k+1) = A265649(n, k) * k! for 0 <= k <= n. (End)

A134144 A certain partition array in Abramowitz-Stegun order (A-St order).

Original entry on oeis.org

1, 3, 1, 15, 9, 1, 105, 60, 27, 18, 1, 945, 525, 450, 150, 135, 30, 1, 10395, 5670, 4725, 2250, 1575, 2700, 405, 300, 405, 45, 1, 135135, 72765, 59535, 55125, 19845, 33075, 15750, 14175, 3675, 9450, 2835, 525, 945, 63, 1, 2027025, 1081080, 873180, 793800

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(3), the k=3 member of a family of generalizations of the multinomial number array M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(3,n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3= A036040.
a(n,k) enumerates unordered forests of increasing ternary trees related to the k-th partition of n in the A-St order. The forest is composed of m such trees, with m the number of parts of the partition.

			[1]; [3,1]; [15,9,1]; [105,60,27,18,1]; [945,525,450,150,135,30,1]; ...
a(4,3)=27 from the partition (2^2) of 4: 4!*((3/2!)^2)/2! = 27.
There are a(4,3) = 27 = 3*3^2 unordered 2-forests with 4 vertices, composed of two increasing ternary trees, each with two vertices: there are 3 increasing labelings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in three versions from the ternary structure.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A049118 (row sums, identical with those of triangle A035342).

a(n,k) = n!*Product_{j=1..n} (S2(3,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(3,n,1) = A035342(n,1) = A001147(n) = (2*n-1)!! and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

A134146 Triangle of numbers obtained from the partition array A134145.

Original entry on oeis.org

1, 3, 1, 15, 3, 1, 105, 24, 3, 1, 945, 150, 24, 3, 1, 10395, 1485, 177, 24, 3, 1, 135135, 14805, 1620, 177, 24, 3, 1, 2027025, 191520, 16425, 1701, 177, 24, 3, 1, 34459425, 2687580, 208125, 16830, 1701, 177, 24, 3, 1, 654729075, 44552025, 2880360, 212985

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(3)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1]; [3,1]; [15,3,1]; [105,24,3,1]; [945,150,24,3,1];...

W. Lang, First 10 rows and more.

Cf. A134147 (row sums).
Cf. A134148 (allternating row sums).
Cf. A134134 (k=2 member of this triangle family).

a(n,m)=sum(product(S2(3;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S2(3;j,1)= A001147(j) = A035342(j,1) = (2*j-1)!!.

A144267 Partition number array, called M32(-4), related to A011801(n,m)= |S2(-4;n,m)| ( generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 36, 12, 1, 504, 144, 48, 24, 1, 9576, 2520, 1440, 360, 240, 40, 1, 229824, 57456, 30240, 12960, 7560, 8640, 960, 720, 720, 60, 1, 6664896, 1608768, 804384, 635040, 201096, 211680, 90720, 60480, 17640, 30240, 6720, 1260, 1680, 84, 1, 226606464, 53319168

Offset: 1

Wolfdieter Lang, Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-4;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+3)-ary trees if the outdegree is r >= 0.
If M32(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle A011801(n,m)= |S2(-4;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

			a(4,3)=48. The relevant partition of 4 is (2^2). The 48 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are quaternary because r=1 vertices are quaternary (4-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4^2=16 versions due to the two quaternary root vertices.

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A143173 (M32(-3) array), A144268 (M32(-5) array).

a(n,k) = (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-4,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-4,j,1)|^e(n,k,j),j=1..n), with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

A113278 Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where [T^2](n,n) = 1 and [T^2](n+1,n) = 2*(n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 3, -3, 3, 1, -15, 12, -6, 4, 1, 105, -75, 30, -10, 5, 1, -945, 630, -225, 60, -15, 6, 1, 10395, -6615, 2205, -525, 105, -21, 7, 1, -135135, 83160, -26460, 5880, -1050, 168, -28, 8, 1, 2027025, -1216215, 374220, -79380, 13230, -1890, 252, -36, 9, 1

Offset: 0

Paul D. Hanna, Oct 22 2005

			Triangle begins:
  1;
  1,1;
  -1,2,1;
  3,-3,3,1;
  -15,12,-6,4,1;
  105,-75,30,-10,5,1;
  -945,630,-225,60,-15,6,1;
  10395,-6615,2205,-525,105,-21,7,1;
  ...
where T(n,k) = (-1)^(n-1-k)*A001147(n-1)*C(n,k).
The matrix square equals:
  1;
  2,1;
  0,4,1;
  0,0,6,1;
  0,0,0,8,1;
  0,0,0,0,10,1;
  0,0,0,0,0,12,1;
  ...
The matrix log, L, begins:
  0;
  1,0;
  -2,2,0;
  8,-6,3,0;
  -48,32,-12,4,0;
  384,-240,80,-20,5,0;
  -3840,2304,-720,160,-30,6,0;
  ...
where L(n,k) = (-1)^(n-1-k)*A000165(n-1)*C(n,k).

Peter Bala, Generalized Dobinski formulas

Cf. A039683, A122848.

Cf. A001147 (odd double factorials), A000165 (even double factorials).

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Sqrt[1 + 2 #]&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r==c,1,if(r==c+1,2*c)))); (sum(i=0,n+1,(sum(j=1,n+1,-(M^0-M)^j/j)/2)^i/i!))[n+1,k+1]}

Exponential Riordan array [sqrt(1 + 2*x),x] with e.g.f. sqrt(1+2*x)*exp(t*x) = 1 + (1+t)*x + (-1+2*t+t^2)*x^2/2! + ... . The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*sum {k = 0..inf} (2*k+1)*(2*k-1)*...*(2*k+1-2*(n-1))*(x/2)^k/k!. Cf. A122848. - Peter Bala, Jun 23 2014

A134145 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.

Original entry on oeis.org

1, 3, 1, 15, 3, 1, 105, 15, 9, 3, 1, 945, 105, 45, 15, 9, 3, 1, 10395, 945, 315, 225, 105, 45, 27, 15, 9, 3, 1, 135135, 10395, 2835, 1575, 945, 315, 225, 135, 105, 45, 27, 15, 9, 3, 1, 2027025, 135135, 31185, 14175, 11025, 10395, 2835, 1575, 945, 675, 945, 315, 225

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(3) = A134144 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(3)/M_3.

			[1]; [3,1]; [15,3,1]; [105,15,9,3,1]; [945,105,45,15,9,3,1]; ...
a(4,3)=9 from the third (k=3) partition (2^2) of 4: (3)^2 = 9, because S2(3,2,1) = 3!! = 1*3 = 3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134147 (row sums, also of triangle A134146).

a(n,k) = Product_{j=1..n} S2(3,j,1)^e(n,k,j) with S2(3,n,1) = A035342(n,1) = A001147(n) = (2*n-1)!! and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134144(n,k)/A036040(n,k) (division of partition arrays M_3(3) by M_3).

A143171 Partition number array, called M32(-1), related to A001497(n-1,m-1) = |S2(-1;n,m)| (generalized Stirling2 triangle).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 12, 3, 6, 1, 105, 75, 30, 30, 15, 10, 1, 945, 630, 225, 90, 225, 180, 15, 60, 45, 15, 1, 10395, 6615, 2205, 1575, 2205, 1575, 630, 315, 525, 630, 105, 105, 105, 21, 1, 135135, 83160, 26460, 17640, 7875, 26460, 17640, 12600, 3150, 2520, 5880, 6300

Offset: 1

Wolfdieter Lang, Oct 09 2008, Dec 04 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-1;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk :=(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = Sum_{j=1..n} e(n,k,j). The special (enk)-forest is composed of m rooted increasing r-ary trees if the outdegree is r >= 0.
This generalizes the array of multinomials called M_3 in Abramowitz-Stegun, pp. 831-2. M_3 = A036040.
If M32(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle A001497(n-1,m-1) = |S2(-1;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

			a(4,3) = 3. The relevant partition of 4 is (2^2). The 3 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing unary trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are unary because r=1 vertices are unary (1-ary) and for the leaves (r=0) the arity does not matter.

Wolfdieter Lang, First 10 rows of the array and more.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A143173 M32(-2) array.

a(n,k) = (n!/Product_{j=1..n} e(n,k,j)!*j!^e(n,k,j)) * Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j) = M3(n,k)*Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j), with |S2(-1,n,1)| = A001147(n-1) = (2*n-3)(!^2) (2-factorials) for n >= 2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k) := A036040(n,k), k=1..p(n), p(n) := A000041(n).

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A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

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A134144 A certain partition array in Abramowitz-Stegun order (A-St order).

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A134146 Triangle of numbers obtained from the partition array A134145.

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