A269941 -id:A269941 - cp's OEIS frontend

A269939 Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.

1, 0, 1, 0, 1, 3, 0, 1, 10, 15, 0, 1, 25, 105, 105, 0, 1, 56, 490, 1260, 945, 0, 1, 119, 1918, 9450, 17325, 10395, 0, 1, 246, 6825, 56980, 190575, 270270, 135135, 0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025

Offset: 0

Peter Luschny, Mar 26 2016

We propose to call this sequence the 'Ward set numbers' and sequence A269940 the 'Ward cycle numbers'. - Peter Luschny, Nov 25 2022

			Triangle starts:
  1;
  0, 1;
  0, 1,   3;
  0, 1,  10,   15;
  0, 1,  25,  105,   105;
  0, 1,  56,  490,  1260,    945;
  0, 1, 119, 1918,  9450,  17325,  10395;
  0, 1, 246, 6825, 56980, 190575, 270270, 135135;

Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
Peter Luschny, The P-transform.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Marko Riedel, Math Stackexchange, Upper Stirling numbers and Ward numbers.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 1, 12.

Variants: A134991 (main entry for this triangle), A181996.
Row sums are A000311.
Alternating row sums are signed factorials A133942.
Cf. A269940 (Stirling1 counterpart), A268437.

# first version
A269939 := (n,k) -> add((-1)^(m+k)*binomial(n+k,n+m)*Stirling2(n+m, m), m=0..k):
seq(seq(A269939(n,k), k=0..n), n=0..8);
# Alternatively:
T := proc(n,k) option remember;
    `if`(k=0 and n=0, 1,
    `if`(k<=0 or k>n, 0,
    k*T(n-1,k)+(n+k-1)*T(n-1,k-1))) end:
for n from 0 to 6 do seq(T(n,k),k=0..n) od;
# simple, third version
T := (n,k)->  (n+k)!*coeftayl((exp(z)-z-1)^k/k!, z=0, n+k); # Marko Riedel, Apr 14 2016

Table[Sum[(-1)^(m + k) Binomial[n + k, n + m] StirlingS2[n + m, m], {m, 0, k}], {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 15 2016 *)

T(n) = {[Vecrev(Pol(p)) | p<-Vec(serlaplace(1/((1+y)*(1 + lambertw(-y/(1+y)*exp((x-y)/(1+y) + O(x*x^n)))))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2022

T = lambda n,k: sum((-1)^(m+k)*binomial(n+k,n+m)*stirling_number2(n+m,m) for m in (0..k))
for n in (0..6): print([T(n,k) for k in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(n+1), lambda n, k: (-1)^k*falling_factorial(n+k,n))

T(n,k) = (-1)^k*FF(n+k,n)*P[n,k](1/(n+1)) where P is the P-transform and FF the falling factorial function. For the definition of the P-transform see the link.
T(n,k) = A268437(n,k)*FF(n+k,n)/(2*n)!.
T(n,k) = (n+k)! [z^{n+k}] (exp(z)-z-1)^k/k!. - Marko Riedel, Apr 14 2016
From Fabián Pereyra, Jan 12 2022: (Start)
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) for n > 0, T(0,0) = 1, T(n,0) = 0 for n > 0. (See the second Maple program.)
E.g.f.: A(x,t) = 1/((1+t)*(1 + W(-t/(1+t)*exp((x-t)/(1+t))))), where W(x) is the Lambert W-function.
T(n,k) = Sum_{j=0..k} E2(n,j)*binomial(n-j,k-j), where E2(n,k) are the second-order Eulerian numbers A340556.
T(n,k) = Sum_{j=k..n} (-1)^(n-j)*A112486(n,j)*binomial(j,k). (End)

A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 21, 14, 1, 0, 1, 85, 147, 30, 1, 0, 1, 341, 1408, 627, 55, 1, 0, 1, 1365, 13013, 11440, 2002, 91, 1, 0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers T(2*n, 2*k) (cf. A036969).

The analog for the Stirling cycle numbers is A269944.

			Triangle starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,   1]
  [3] [0, 1,   5,    1]
  [4] [0, 1,  21,   14,   1]
  [5] [0, 1,  85,  147,  30,  1]
  [6] [0, 1, 341, 1408, 627, 55, 1]

M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.

Columns k=0..5 give A000007, A000012, A002450(n-1), A002451(n-3), A383838(n-4), A383840(n-5).
Variants are: A008957, A036969.
Cf. A007318 (order 0), A048993 (order 1), A269948 (order 3).
Cf. A000330 (subdiagonal), A002450 (column 2), A135920 (row sums), A269941, A269944 (Stirling cycle), A298851 (central terms).

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od;
# Alternatively with the P-transform (cf. A269941):
A269945_row := n -> PTrans(n, n->`if`(n=1, 1, 1/(n*(4*n-2))), (n, k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269945_row(n)), n=0..8);
# Using the exponential generating function:
egf := 1 + t^2*(cosh(2*sinh(t*x/2)/t));
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, 2*(n-k+1)), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # Peter Luschny, Feb 29 2024

T[n_, n_] = 1; T[n_ /; n >= 0, k_] /; 0 <= k < n := T[n, k] = T[n - 1, k - 1] + k^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Nov 27 2017 *)

# uses[PtransMatrix from A269941]
stirset2 = lambda n: 1 if n == 1 else 1/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stirset2, norm)
for m in M: print(m)

T(n, k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = 1/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n, 2) = (4^(n - 1) - 1)/3 for n >= 2 (cf. A002450).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n, k) = (1/(2*k)!)*Sum_{j=0..2*k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n).
T(n, k) = Sum_{j=2*k..2*n} (-k)^(2*n - j)*binomial(2*n, j)*Stirling2(j, 2*k).
T(n, k) = Sum_{j=0..2*n} (-1)^(j - k)*Stirling2(2*n - j, k)*Stirling2(j, k). (End)
T(n, k) = (2*n)! [t^(2*(n-k+1))] [x^(2*n)] (1 + t^2*(cosh(2*sinh(t*x/2)/t))). - Peter Luschny, Feb 29 2024

A268437 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](1/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 4, 6, 0, 30, 120, 90, 0, 336, 2800, 5040, 2520, 0, 5040, 80640, 264600, 302400, 113400, 0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400, 0, 2162160, 118198080, 983782800, 2986663680, 4162158000, 2724321600, 681080400

Offset: 0

Peter Luschny, Mar 07 2016

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

			[1],
[0, 1],
[0, 4, 6],
[0, 30, 120, 90],
[0, 336, 2800, 5040, 2520],
[0, 5040, 80640, 264600, 302400, 113400],
[0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400].

Peter Luschny, The P-transform.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Cf. A000680, A001761, A268438, A269939, A269941.

A268437 := proc(n,k) local F,T;
  F := proc(n,k) option remember;
  `if`(n=0 and k=0, 1,`if`(n=k, (4*n-2)*F(n-1,k-1),
  F(n-1,k)*(n+k))) end;
  T := proc(n,k) option remember;
  `if`(k=0 and n=0, 1,`if`(k<=0 or k>n, 0,
  (4*n-2)*n*(k*T(n-1,k)+(n+k-1)*T(n-1,k-1)))) end;
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268437(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268437_row := n -> PTrans(n, n->1/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(print(A268437_row(n)),n=0..8);

T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] StirlingS2[n+m, m], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)

A268437 = lambda n, k: (factorial(2*n)/falling_factorial(n+k, n))*sum((-1)^(m+k)* binomial(n+k, n+m)*stirling_number2(n+m, m) for m in (0..k))
for n in (0..7): print([A268437(n, m) for m in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(n+1), lambda n, k: (-1)^k* factorial(2*n))

T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling2(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269939(n,k).
T(n,1) = (2*n)!/(n+1)! = A001761(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.

A268438 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 8, 6, 0, 180, 240, 90, 0, 8064, 14560, 10080, 2520, 0, 604800, 1330560, 1285200, 604800, 113400, 0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400, 0, 10897286400, 30858347520, 43770767040, 36970053120, 18918900000, 5448643200, 681080400

Offset: 0

Peter Luschny, Mar 07 2016

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

			Triangle starts:
[1],
[0, 1],
[0, 8,        6],
[0, 180,      240,       90],
[0, 8064,     14560,     10080,     2520],
[0, 604800,   1330560,   1285200,   604800,    113400],
[0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400].

Peter Luschny, The P-transform.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Cf. A000680, A060593, A268437, A269940, A269941.

A268438 := proc(n,k) local F,T;
  F := proc(n,k) option remember;
  `if`(n=0 and k=0, 1,`if`(n=k, (4*n-2)*F(n-1,k-1),
  F(n-1,k)*(n+k))) end;
  T := proc(n, k) option remember;
  `if`(k=0 and n=0, 1,`if`(k<=0 or k>n, 0,
  (4*n-2)*n*(n+k-1)*(T(n-1,k)+T(n-1,k-1)))) end:
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268438(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268438_row := n -> PTrans(n, n->n/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(lprint(A268438_row(n)), n=0..8);

T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)

A268438 = lambda n,k: (factorial(2*n)/falling_factorial(n+k,n))*sum((-1)^(m+k)* binomial(n+k,n+m)*stirling_number1(n+m,m) for m in (0..k))
for n in (0..7): print([A268438(n,m) for m in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n))

T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling1(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269940(n,k).
T(n,1) = (2*n)!/(n+1) = A060593(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.

A269944 Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 5, 1, 0, 36, 49, 14, 1, 0, 576, 820, 273, 30, 1, 0, 14400, 21076, 7645, 1023, 55, 1, 0, 518400, 773136, 296296, 44473, 3003, 91, 1, 0, 25401600, 38402064, 15291640, 2475473, 191620, 7462, 140, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers |t(2*n, 2*k)| (cf. A008955).

The analog for the Stirling set numbers is A269945.

			Triangle starts:
  [1]
  [0,     1]
  [0,     1,     1]
  [0,     4,     5,    1]
  [0,    36,    49,   14,    1]
  [0,   576,   820,  273,   30,  1]
  [0, 14400, 21076, 7645, 1023, 55, 1]

Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.
B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.

Variants: A204579 (signed, row 0 missing), A008955.
Cf. A007318 (order 0), A132393 (order 1), A269947 (order 3).
Cf. A000330 (subdiagonal), A001044 (column 1), A101686 (row sums), A269945 (Stirling set), A269941 (P-transform).

T := proc(n, k) option remember; if n=k then return 1 fi; if k<0 or k>n then return 0 fi; T(n-1, k-1)+(n-1)^2*T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A269944_row := n -> PTrans(n, n->`if`(n=1, 1, (n-1)^2/(n*(4*n-2))), (n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269944_row(n)), n=0..8);
# From Peter Luschny, Feb 29 2024: (Start)
# Computed as the coefficients of polynomials:
P := (x, n) -> local j; mul((j - x)*(j + x), j = 0..n-1):
T := (n, k) -> (-1)^k*coeff(P(x, n), x, 2*k):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
# Alternative, using the exponential generating function:
egf := cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # (End)
# Alternative, row polynomials:
rowpoly := n -> pochhammer(-sqrt(x), n) * pochhammer(sqrt(x), n):
row := n -> local k; seq((-1)^k*coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..6);  # Peter Luschny, Aug 03 2024

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + (n - 1)^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Jul 25 2019 *)

stircycle2 = lambda n: 1 if n == 1 else (n-1)^2/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stircycle2, norm)
for m in M: print(m)

T(n,k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = (n - 1)^2 / (n*(4*n - 2)). The P-transform is defined in the link. See the Sage and Maple implementations below.
T(n, 1) = ((n - 1)!)^2 for n >= 1 (cf. A001044).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
Row sums: Product_{k=1..n} ((k - 1)^2 + 1) for n >= 0 (cf. A101686).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n,k) = (-1)^(n-k)*Sum_{j=2*k..2*n} Stirling1(2*n,j)*binomial(j,2*k)*(n-1)^(j-2*k).
T(n,k) = Sum_{j=0..2*k} (-1)^(j - k)*Stirling1(n, j)*Stirling1(n, 2*k - j). (End)
From Peter Luschny, Feb 29 2024: (Start)
T(n, k) = (-1)^k*[x^(2*k)] P(x, n) where P(x, n) = Product_{j=0..n-1} (j-x)*(j+x).
T(n, k) = (2*n)!*[t^(n-k)] [x^(2*n)] cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)). (End)
T(n, k) = (-1)^k*[x^k] Pochhammer(-sqrt(x), n) * Pochhammer(sqrt(x), n). - Peter Luschny, Aug 03 2024

A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)binomial(n + k, n + m) |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 6, 20, 15, 0, 24, 130, 210, 105, 0, 120, 924, 2380, 2520, 945, 0, 720, 7308, 26432, 44100, 34650, 10395, 0, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025

Offset: 0

Peter Luschny, Mar 27 2016

We propose to call this sequence the 'Ward cycle numbers' and sequence A269939 the 'Ward set numbers'. - Peter Luschny, Nov 25 2022

			Triangle T(n,k) starts:
  [1]
  [0,   1]
  [0,   2,      3]
  [0,   6,     20,     15]
  [0,  24,    130,    210,    105]
  [0,  120,   924,   2380,   2520,    945]
  [0,  720,  7308,  26432,  44100,  34650,  10395]
  [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135]

Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
Peter Luschny, The P-transform.
Anthony Mansuy, Preordered forests, packed words and contraction algebras, J. Algebra 411 (2014) 259-311, section 4.4.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
Nico M. Temme, Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters, Integral Transforms and Special Functions, 2021.
Nico M. Temme and Ed J. M. Veling, Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z, arXiv:2202.12857 [math.CA], 2022.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Variants: A111999, A259456.

Cf. A269939 (Stirling2 counterpart), A268438, A032188 (row sums).

T := (n, k) -> add((-1)^(m+k)*binomial(n+k,n+m)*abs(Stirling1(n+m, m)), m=0..k):
seq(print(seq(T(n, k), k=0..n)), n=0..6);
# Alternatively:
T := proc(n, k) option remember;
    `if`(k=0, k^n,
    `if`(k<=0 or k>n, 0,
    (n+k-1)*(T(n-1, k)+T(n-1, k-1)))) end:
for n from 0 to 6 do seq(T(n, k), k=0..n) od;

T[n_, k_] := Sum[(-1)^(m+k)*Binomial[n+k, n+m]*Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 12 2022 *)

T = lambda n, k: sum((-1)^(m+k)*binomial(n+k, n+m)*stirling_number1(n+m, m) for m in (0..k))
for n in (0..7): print([T(n, k) for k in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: n/(n+1), lambda n, k: (-1)^k*falling_factorial(n+k,n))

T(n,k) = (-1)^k*FF(n+k,n)*P[n,k](n/(n+1)) where P is the P-transform and FF the falling factorial function. For the definition of the P-transform see the link.

T(n,k) = A268438(n,k)*FF(n+k,n)/(2*n)!.

Name corrected after notice from Ed Veling by Peter Luschny, Jun 14 2022

A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 10, 1, 0, 100, 140, 28, 1, 0, 1700, 2900, 840, 60, 1, 0, 44200, 85800, 31460, 3300, 110, 1, 0, 1635400, 3476200, 1501500, 203060, 10010, 182, 1, 0, 81770000, 185874000, 90563200, 14700400, 943800, 25480, 280, 1

Offset: 0

Peter Luschny, Mar 07 2016

0

			[1]
[0,        1]
[0,        2,         1]
[0,       10,        10,        1]
[0,      100,       140,       28,        1]
[0,     1700,      2900,      840,       60,      1]
[0,    44200,     85800,    31460,     3300,    110,     1]
[0,  1635400,   3476200,  1501500,   203060,  10010,   182,   1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A269946 (order 3).

Cf. A036969, A105278, A204579, A269938, A269941, A269944, A269945.

T := proc(n,k) option remember;
if n=k then return 1 fi; if k<0 or k>n then return 0 fi;
T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k) end:
seq(seq(T(n,k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A268434_row := n -> PTrans(n, n->`if`(n=1,1, ((n-1)^2+1)/(n*(4*n-2))),
(n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A268434_row(n)), n=0..8);

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^2 + k^2)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

#cached_function
def T(n, k):
    if n==k: return 1
    if k<0 or k>n: return 0
    return T(n-1, k-1)+((n-1)^2+k^2)*T(n-1, k)
for n in range(8): print([T(n, k) for k in (0..n)])
# Alternatively with the function PtransMatrix (cf. A269941):
PtransMatrix(8, lambda n: 1 if n==1 else ((n-1)^2+1)/(n*(4*n-2)), lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k))

T(n,k) = (-1)^k*((2*n)!/(2*k)!)*P[n,k](s(n)) where P is the P-transform and s(n) = ((n-1)^2+1)/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n,k) = Sum_{j=k..n} A269944(n,j)*A269945(j,k).
T(n,1) = Product_{k=1..n} (k-1)^2+1 for n>=1 (cf. A101686).
T(n,n-1) = (n-1)*n*(2*n-1)/3 for n>=1 (cf. A006331).
Row sums: A269938.

A268439 Triangle read by rows, T(n,k) = C(2n,n+k)Sum_{m=0..k} (-1)^(m+k)C(n+k,n+m) Stirling2(n+m,m), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 4, 3, 0, 15, 60, 15, 0, 56, 700, 840, 105, 0, 210, 6720, 22050, 12600, 945, 0, 792, 58905, 421960, 623700, 207900, 10395, 0, 3003, 492492, 6831825, 20740720, 17342325, 3783780, 135135, 0, 11440, 4012008, 100180080, 551450900, 916515600, 491891400, 75675600, 2027025

Offset: 0

Peter Luschny, Mar 08 2016

			[1]
[0, 1]
[0, 4, 3]
[0, 15, 60, 15]
[0, 56, 700, 840, 105]
[0, 210, 6720, 22050, 12600, 945]
[0, 792, 58905, 421960, 623700, 207900, 10395]
[0, 3003, 492492, 6831825, 20740720, 17342325, 3783780, 135135]

Peter Luschny, The P-transform.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Cf. A001147, A001791, A268437, A268440, A269939, A269941.

# The function PTrans is defined in A269941.
A268439_row := n -> PTrans(n, n->1/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)):
seq(print(A268439_row(n)), n=0..8);

A268439 = lambda n, k: binomial(2*n, n+k)*sum((-1)^(m+k)*binomial(n+k, n+m)* stirling_number2(n+m, m) for m in (0..k))
for n in (0..7): print([A268439(n, m) for m in (0..n)])

# uses[PtransMatrix from A269941]
# Alternatively
PtransMatrix(8, lambda n: 1/(n+1), lambda n,k: (-1)^k*factorial(2*n)/ (factorial(k)*factorial(n-k)))

T(n,k) = ((-1)^k*(2*n)!/(k!*(n-k)!))*P[n,k](1/(n+1)) where P is the P-transform. The P-transform is defined in the link.
T(n,k) = A269939(n,k)*binomial(2*n,n+k).
T(n,k) = A268437(n,k)/(k!*(n-k)!).
T(n,1) = binomial(2*n,n-1) = A001791(n) for n>=1.
T(n,n) = (2*n-1)!! = A001147(n) for n>=0.

A268440 Triangle read by rows, T(n,k) = C(2n,n+k)Sum_{m=0..k} (-1)^(m+k)C(n+k,n+m) Stirling1(n+m,m), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 8, 3, 0, 90, 120, 15, 0, 1344, 3640, 1680, 105, 0, 25200, 110880, 107100, 25200, 945, 0, 570240, 3617460, 5815040, 2910600, 415800, 10395, 0, 15135120, 128576448, 303963660, 256736480, 78828750, 7567560, 135135

Offset: 0

Peter Luschny, Mar 08 2016

			[1]
[0, 1]
[0, 8, 3]
[0, 90, 120, 15]
[0, 1344, 3640, 1680, 105]
[0, 25200, 110880, 107100, 25200, 945]
[0, 570240, 3617460, 5815040, 2910600, 415800, 10395]

Peter Luschny, The P-transform.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Cf. A001147, A092956, A268438, A268439, A269940, A269941.

# The function PTrans is defined in A269941.
A268440_row := n -> PTrans(n, n->n/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)):
seq(print(A268440_row(n)), n=0..8);

A268440 = lambda n, k: binomial(2*n,n+k)*sum((-1)^(m+k)*binomial(n+k,n+m)* stirling_number1(n+m, m) for m in (0..k))
for n in (0..7): print([A268440(n, m) for m in (0..n)])

# uses[PtransMatrix from A269941]
# Alternatively
PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n)/ (factorial(k)*factorial(n-k)))

T(n,k) = ((-1)^k*(2*n)!/(k!*(n-k)!))*P[n,k](n/(n+1)) where P is the P-transform. The P-transform is defined in the link.
T(n,k) = A269940*binomial(2*n,n+k).
T(n,k) = A268438(n,k)/(k!*(n-k)!).
T(n,1) = n*(2*n)!/(n+1)! for n>=1 (cf. A092956).
T(n,n) = (2*n-1)!! = A001147(n) for n>=0.

A357078 Triangle read by rows. The partition transform of A355488, which are the alternating row sums of the number of permutations of [n] with k components (A059438).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 8, 4, 0, 1, 0, 48, 16, 6, 0, 1, 0, 328, 100, 24, 8, 0, 1, 0, 2560, 688, 156, 32, 10, 0, 1, 0, 22368, 5376, 1080, 216, 40, 12, 0, 1, 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1, 0, 2291456, 450240, 73440, 11808, 1960, 348, 56, 16, 0, 1

Offset: 0

Peter Luschny, Sep 10 2022

The partition transform (also called De Moivre polynomials by Cormac O'Sullivan) is defined in the program section as a Sage script.

The triangle represents a refinement of the number of irreducible permutations, A003319. Together with the refinement of the number of reducible permutations A356265 the triangle sums to the refinement of the factorial numbers given in A357079.

			Triangle T(n, k) starts:                         [Row sums]
[0] 1;                                               [1]
[1] 0,      1;                                       [1]
[2] 0,      0,     1;                                [1]
[3] 0,      2,     0,    1;                          [3]
[4] 0,      8,     4,    0,    1;                    [13]
[5] 0,     48,    16,    6,    0,   1;               [71]
[6] 0,    328,   100,   24,    8,   0,  1;           [461]
[7] 0,   2560,   688,  156,   32,  10,  0,  1;       [3447]
[8] 0,  22368,  5376, 1080,  216,  40, 12,  0, 1;    [29093]
[9] 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1; [273343]

Peter Luschny, The P-transform.

Cf. A355488, A003319, A059438, A356265, A357079, A269941.

from functools import cache
def PartTrans(dim, f):
    X = var(['x' + str(i) for i in range(dim + 1)])
    @cache
    def PCoeffs(n: int, k: int):
        R = PolynomialRing(ZZ, X[1: n - k + 2], n - k + 1, order='lex')
        if k == 0: return R(k^n)
        return R(sum(PCoeffs(n - j, k - 1) * f(j)
                     for j in range(1, n - k + 2)))
    return [[PCoeffs(n, k) for k in range(n + 1)] for n in range(dim)]
def A357078_triangle(dim):
    A = ZZ[['t']]; g = A([0] + [factorial(n) for n in range(1, 30)]).O(dim+2)
    return PartTrans(dim, lambda n: list(g / (1 + 2 * g))[n])
A357078_triangle(9)

cp's OEIS Frontend

A269939 Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.

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A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

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A268437 Triangle read by rows, T(n,k) = (-1)^k*(2*n)!*P[n,k](1/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

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A268438 Triangle read by rows, T(n,k) = (-1)^k*(2*n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

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A269944 Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

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A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)*binomial(n + k, n + m) * |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

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A268437 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](1/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

A268438 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)binomial(n + k, n + m) |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

A268439 Triangle read by rows, T(n,k) = C(2n,n+k)Sum_{m=0..k} (-1)^(m+k)C(n+k,n+m) Stirling2(n+m,m), for n>=0 and 0<=k<=n.

A268440 Triangle read by rows, T(n,k) = C(2n,n+k)Sum_{m=0..k} (-1)^(m+k)C(n+k,n+m) Stirling1(n+m,m), for n>=0 and 0<=k<=n.