A035342 -id:A035342 - cp's OEIS frontend

A143172 Partition number array, called M32(-2), related to A004747(n,m) = |S2(-2;n,m)| (generalized Stirling triangle).

1, 2, 1, 10, 6, 1, 80, 40, 12, 12, 1, 880, 400, 200, 100, 60, 20, 1, 12320, 5280, 2400, 1000, 1200, 1200, 120, 200, 180, 30, 1, 209440, 86240, 36960, 28000, 18480, 16800, 7000, 4200, 2800, 4200, 840, 350, 420, 42, 1, 4188800, 1675520, 689920, 492800, 224000, 344960

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(-2;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+1)-ary trees if the outdegree is r>=0.
If M32(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle A004747(n,m)= |S2(-2;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)=12. 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 binary because r=1 vertices are binary (2-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 binary 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. A143171 (M32(-1) array), A143173 (M32(-3) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-2,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-2,j,1)|^e(n,k,j),j=1..n), with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-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).

A144268 Partition number array, called M32(-5), related to A013988(n,m)= |S2(-5;n,m)| ( generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 55, 15, 1, 935, 220, 75, 30, 1, 21505, 4675, 2750, 550, 375, 50, 1, 623645, 129030, 70125, 30250, 14025, 16500, 1875, 1100, 1125, 75, 1, 21827575, 4365515, 2258025, 1799875, 451605, 490875, 211750, 144375, 32725, 57750, 13125, 1925, 2625, 105, 1, 894930575

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(-5;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+4)-ary trees if the outdegree is r >= 0.
If M32(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle A013988(n,m)= |S2(-5;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)=75. The relevant partition of 4 is (2^2). The 75 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 5-ary because r=1 vertices are 5-ary and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 5^2=25 versions due to the two 5-ary 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. A144267 (M32(-4) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-5,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-5,j,1)|^e(n,k,j),j=1..n), with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-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).

A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n is odd, and of 3^(n/2)(x^(2/3)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 3, 4, 3, 4, 24, 9, 28, 42, 9, 28, 252, 189, 27, 280, 630, 270, 27, 280, 3360, 3780, 1080, 81, 3640, 10920, 7020, 1404, 81, 3640, 54600, 81900, 35100, 5265, 243, 58240, 218400, 187200, 56160, 6480, 243, 58240, 1048320, 1965600

Offset: 0

Udita Katugampola, Mar 18 2013

			Triangle begins:
1;
1, 3;
4, 3;
4, 24, 9;
28, 42, 9;
28, 252, 189, 27;
280, 630, 270, 27;
280, 3360, 3780, 1080, 81;
3640, 10920, 7020, 1404, 81;
3640, 54600, 81900, 35100, 5265, 243,
58240, 218400, 187200, 56160, 6480, 243

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
end do;

A035119 Related to A045720 and A035101.

Original entry on oeis.org

0, 0, 1, 18, 285, 4680, 82845, 1595790, 33453945, 760970700, 18705542625, 494764058250, 14023390706325, 424278354099600, 13653335491921125, 465794724725079750, 16796514560465264625, 638448710154151396500

Offset: 1

Wolfdieter Lang

3rd column of triangular array A035342. a(n) = (2*n+1)*a(n-1) + A035101(n-1), n >= 3, a(2)=0.
a(n) gives the number of organically labeled forests (sets) with three rooted ordered trees with n non-root vertices. 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. W. Lang, Aug 07 2007.
a(n), n>=3, enumerates unordered n-vertex forests composed of three 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(4)=18 for the number of forests (sets) of three increasing labeled rooted trees with 4 non-root vertices and three root labels 0: [(0,4),{(0,1),(0,2)},(0,3)]; [(0,4),{(0,2),(0,1)},(0,3)]; [(0,4),{(0,1),(0,3)},(0,2)]; [(0,4),{(0,3),(0,1)},(0,2)]; [(0,4),{(0,2),(0,3)},(0,1)]; [(0,4),{(0,3),(0,2)},(0,1)]; [(0,4),(0,1,2),(0,3)]; [(0,4),(0,1,3),(0,2)]; [(0,4),(0,2,3),(0,1)]; [{(0,4),(0,1)},(0,2),(0,3)]; [{(0,1),(0,4)},(0,2),(0,3)]; [{(0,4),(0,2)},(0,1),(0,3)]; [{(0,2),(0,4)},(0,1),(0,3)]; [{(0,4),(0,3)},(0,1),(0,2)]; [{(0,3),(0,4)},(0,1),(0,2)]; [(0,1,4),(0,2),(0,3)]; [(0,2,4),(0,1),(0,3)]; [(0,3,4),(0,1),(0,2)].
a(4)=18 increasing ternary 3-forest with n=4 vertices: there are three 3-forests (two one vertex trees together with any of the three different 2-vertex trees) each with six increasing labelings. W. Lang, Sep 14 2007.

Cf. A000108, A045720, A035101, A035342.

a(n) = n!*((n+2)*binomial(2*n, n)/4-3*2^(2*n-3))/(3*2^(n-2)); a(n)= n!*A045720(n-3)/(3*2^(n-2)), n >= 3; E.g.f. (4/3)*(x*c(x/2)*(1-2*x)^(-1/2)/2)^3 = (2*x/3)*((1-x/2)*c(x/2)-1)/(1-2*x)^(3/2), where c(x) = g.f. for Catalan numbers A000108, a(0) := 0.

A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)(x^(1/4)d/dx)^n when n is odd, and of 4^(n/2)(x^(3/4)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 4, 5, 4, 5, 40, 16, 45, 72, 16, 45, 540, 432, 64, 585, 1404, 624, 64, 585, 9360, 11232, 3328, 256, 9945, 31824, 21216, 4352, 256, 9945, 198900, 318240, 141440, 21760, 1024, 208845, 835380, 742560, 228480, 26880, 1024, 208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096

Offset: 0

Udita Katugampola, Mar 20 2013

			Triangle begins:
1;
1, 4;
5, 4;
5, 40, 16;
45, 72, 16;
45, 540, 432, 64;
585, 1404, 624, 64;
585, 9360, 11232, 3328, 256;
9945, 31824, 21216, 4352, 256;
9945, 198900, 318240, 141440, 21760, 1024;
208845, 835380, 742560, 228480, 26880, 1024;
208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(4^((i+1)mod 2)*x^((2((i+1)mod 2)+1)/4)*(diff(a[i-1],x$1 )));
end do;

nmax = 12;
b[0] = Exp[x]; For[ i = 1 , i <= nmax , i++, b[i] = 4^Mod[i + 1, 2]*x^((2 Mod[i + 1, 2] + 1)/4)*D[b[i - 1], x]] // Simplify;
row[1] = {1}; row[n_] := List @@ Expand[b[n]/f[x]] /. x -> 1;
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Feb 22 2019, from Maple *)

Missing terms inserted by Jean-François Alcover, Feb 22 2019

A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)(x^(1/5)d/dx)^n when n is odd, and of 5^(n/2)(x^(4/5)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 5, 6, 5, 6, 60, 25, 66, 110, 25, 66, 990, 825, 125, 1056, 2640, 1200, 125, 1056, 21120, 26400, 8000, 625, 22176, 73920, 50400, 10500, 625, 22176, 554400, 924000, 420000, 65625, 3125, 576576, 2402400, 2184000, 682500, 81250, 3125, 576576, 17297280

Offset: 0

Udita Katugampola, Mar 20 2013

			Triangle begins:
1;
1, 5;
6, 5;
6, 60, 25;
66, 110, 25;
66, 990, 825, 125;
1056, 2640, 1200, 125;
1056, 21120, 26400, 8000, 625;
22176, 73920, 50400, 10500, 625;
22176, 554400, 924000, 420000, 65625, 3125;
576576, 2402400, 2184000, 682500, 81250, 3125;
576576, 17297280, 36036000, 21840000, 5118750, 487500, 15625;
17873856, 89369280, 101556000, 42315000, 7556250, 581250, 15625;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(5^((i+1)mod 2)*x^((3((i+1)mod 2)+1)/5)*(diff(a[i-1],x$1 )));
end do;

A265649 Triangle of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..n} T(n,k)x^k where T(0,0) = 1, and T(n,k) = 0 for k < 0 or k > n, and T(n,k) = T(n-1,k-1) + (2n-1+k)*T(n-1,k) for n > 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 5, 1, 15, 33, 12, 1, 105, 279, 141, 22, 1, 945, 2895, 1830, 405, 35, 1, 10395, 35685, 26685, 7500, 930, 51, 1, 135135, 509985, 435960, 146685, 23310, 1848, 70, 1, 2027025, 8294895, 7921305, 3076290, 589575, 60270, 3318, 92, 1, 34459425, 151335135, 158799690, 69447105, 15457365, 1915515, 136584, 5526, 117, 1

Offset: 0

Werner Schulte, Dec 11 2015

The polynomials p(n,x) satisfy the differential equation: x*y''' + (3*x+1)*y'' + (2*x+2)*y' - 2*n*y = 0 where y' = dy/dx (first derivative).

Appears to be the exponential Riordan array [1/sqrt(1 - 2x), 1/(sqrt(1 - 2x) - 1)]. [Barry, Example 1] - Eric M. Schmidt, Sep 23 2017

			The triangle T(n,k) begins:
n\k:        0        1        2        3       4      5     6   7  8
  0:        1
  1:        1        1
  2:        3        5        1
  3:       15       33       12        1
  4:      105      279      141       22       1
  5:      945     2895     1830      405      35      1
  6:    10395    35685    26685     7500     930     51     1
  7:   135135   509985   435960   146685   23310   1848    70   1
  8:  2027025  8294895  7921305  3076290  589575  60270  3318  92  1
  etc.
The polynomial corresponding to row 3 is p(3,x) = 15 + 33*x + 12*x^2 + x^3.

Paul Barry, A Note on d-Hankel Transforms, Continued Fractions, and Riordan Arrays, arXiv:1702.04011 [math.CO], 2017.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See pp. 21, 28.

Cf. A000007, A000326, A001147, A001497, A021009, A129890, A035342.

T := (n, k) -> local j; 2^n*add((-1)^(k-j)*binomial(k, j)*pochhammer((j+1)/2, n), j=0..k) / k!: for n from 0 to 6 do seq(T(n, k), k=0..n) od;  # Peter Luschny, Mar 04 2024

(* The function RiordanArray is defined in A256893. *)
rows = 10;
R = RiordanArray[1/Sqrt[1 - 2 #]&, 1/Sqrt[1 - 2 #] - 1&, rows, True];
R // Flatten (* Jean-François Alcover, Jul 20 2019 *)

Recurrence: p(0,x) = 1 and p(n+1,x) = (2*n+1+x)*p(n,x) + x*p'(n,x).
T(n,0) = A001147(n), T(n+1,1) = A129890(n), T(n+1,n) = A000326(n+1), and Sum_{k=0..n} (-1)^k*k!*T(n,k) = A000007(n).
Recurrence: k^2*(k+1)*T(n,k+1) = (2*n+2-2*k)*T(n,k-1)-k*(3*k-1)*T(n,k).
Conjecture: T(n,k) = 2^(n-k)*(n-k)!*binomial(n,k)*(Sum_{j=0..n-k} (-1/4)^j* binomial(2*j+k,j)*binomial(n,j+k)).
Conjecture: T(n,k) = (-1)^k*Sum_{j=0..n-1} A001497(n-1,j)*A021009(j+1,k).
T(n,k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i)*Product_{j=1..n} (2*j+i-1))/k!. - Werner Schulte, Mar 03 2024
T(n,k) = (2^n/k!)*(Sum_{j=0..k}(-1)^(k-j)*binomial(k,j)*Pochhammer((j + 1)/2, n)). - Peter Luschny, Mar 04 2024

A371080 Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 28, 12, 1, 0, 280, 160, 24, 1, 0, 3640, 2520, 520, 40, 1, 0, 58240, 46480, 11880, 1280, 60, 1, 0, 1106560, 987840, 295960, 40040, 2660, 84, 1, 0, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1

Offset: 0

Peter Luschny, Mar 12 2024

			Triangle starts:
[0] 1;
[1] 0,       1;
[2] 0,       4,      1;
[3] 0,      28,     12,      1;
[4] 0,     280,    160,     24,     1;
[5] 0,    3640,   2520,    520,    40,    1;
[6] 0,   58240,  46480,  11880,  1280,   60,  1;
[7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;

Peter Luschny, The Bell transform.

Variant: A035469.
Cf. A264428, A371076, A371077.
Cf. A007559, A144346.
Cf. A035342, A049029, A271703.

a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
# Alternative:
BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x),x, 22), x, n), 9);
# Recurrence:
T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 13 2024

T(n, k) = sum(j=k, n, 3^(n-j)*abs(stirling(n, j, 1))*stirling(j, k, 2)); \\ Seiichi Manyama, Apr 19 2025

T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
T(n, k) = A371076(n, k) / k!.
From Werner Schulte, Mar 13 2024: (Start)
T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
T(n, k) = T(n-1, k-1) + (3*(n - 1) + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0. (End)
From Seiichi Manyama, Apr 19 2025: (Start)
T(n,k) = Sum_{j=k..n} 3^(n-j) * |Stirling1(n,j)| * Stirling2(j,k).
E.g.f. of column k (with leading zeros): (1/(1 - 3*x)^(1/3) - 1)^k / k!. (End)

A131449 Number of organic (also called increasing) vertex labelings of rooted ordered trees with n non-root vertices.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 3, 2, 1, 24, 12, 12, 12, 8, 8, 6, 6, 4, 4, 3, 3, 2, 1, 120, 60, 60, 60, 60, 40, 40, 40, 30, 30, 30, 30, 30, 24, 20, 20, 20, 20, 20, 15, 15, 15, 15, 12, 12, 12, 10, 10, 10, 10, 8, 8, 6, 6, 5, 5, 4, 4, 3, 3, 2, 1, 720

Offset: 0

Wolfdieter Lang, Aug 07 2007

Organic vertex labeling with numbers 1,2,...,n means that the sequence of vertex labels along the (unique) path from the root with label 0 to any leaf (non-root vertex of degree 1) is increasing.
Row lengths sequence, i.e. the number of rooted ordered trees, C(n):=A000108(n) (Catalan numbers): [1,1,2,5,14,42,...].
Number of rooted trees with n non-root vertices [1,1,2,4,9,20,...]=A000081(n+1).
Row sums give [1,1,3,155,105,945,...]= A001147(n), n>=0. A035342(n,1), n>=1, first column of triangle S2(3).

			[0! ]; [1! ]; [2!,1]; [3!,3,3,2,1], [4!,12,12,12,8,8,6,6,4,4,3,3,2,1];...
n=3: 3 labelings (0,1,2)(0,3), (0,1,3) (0,2) and (0,2,3) (0,1) for the rooted tree o-o-x-o.
n=3: 3 labelings (0,3)(0,1,2), (0,2)(0,1,3) and (0,1)(0,2,3) for the rooted tree o-x-o-o.

W. Lang, First 6 rows.
W. Lang, Rooted ordered trees with n=5 non-root vertices and number of labelings.

A223512 Triangle T(n,k) represents the coefficients of (x^10*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 10, 1, 190, 30, 1, 5320, 1060, 60, 1, 196840, 45600, 3400, 100, 1, 9054640, 2340040, 208800, 8300, 150, 1, 498005200, 140096880, 14241640, 690200, 17150, 210, 1, 31872332800, 9604302400, 1080045120, 60485040, 1856400, 31640, 280, 1, 2326680294400

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
10,1;
190,30,1;
5320,1060,60,1;
196840,45600,3400,100,1;
9054640,2340040,208800,8300,150,1;
498005200,140096880,14241640,690200,17150,210,1;
31872332800,9604302400,1080045120,60485040,1856400,31640,280,1,2326680294400

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

b[0]:=g(x):
for j from 1 to 10 do
b[j]:=simplify(x^10*diff(b[j-1],x$1);
end do;

cp's OEIS Frontend

A143172 Partition number array, called M32(-2), related to A004747(n,m) = |S2(-2;n,m)| (generalized Stirling triangle).

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A144268 Partition number array, called M32(-5), related to A013988(n,m)= |S2(-5;n,m)| ( generalized Stirling triangle).

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A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)*(x^(1/3)*d/dx)^n when n is odd, and of 3^(n/2)*(x^(2/3)*d/dx)^n when n is even.

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Maple

A035119 Related to A045720 and A035101.

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A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n is odd, and of 4^(n/2)*(x^(3/4)*d/dx)^n when n is even.

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A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)*(x^(1/5)*d/dx)^n when n is odd, and of 5^(n/2)*(x^(4/5)*d/dx)^n when n is even.

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A265649 Triangle of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^k where T(0,0) = 1, and T(n,k) = 0 for k < 0 or k > n, and T(n,k) = T(n-1,k-1) + (2*n-1+k)*T(n-1,k) for n > 0 and 0 <= k <= n.

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A371080 Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

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A131449 Number of organic (also called increasing) vertex labelings of rooted ordered trees with n non-root vertices.

A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n is odd, and of 3^(n/2)(x^(2/3)d/dx)^n when n is even.

A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)(x^(1/4)d/dx)^n when n is odd, and of 4^(n/2)(x^(3/4)d/dx)^n when n is even.

A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)(x^(1/5)d/dx)^n when n is odd, and of 5^(n/2)(x^(4/5)d/dx)^n when n is even.

A265649 Triangle of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..n} T(n,k)x^k where T(0,0) = 1, and T(n,k) = 0 for k < 0 or k > n, and T(n,k) = T(n-1,k-1) + (2n-1+k)*T(n-1,k) for n > 0 and 0 <= k <= n.