A269940 -id:A269940 - cp's OEIS frontend

A269957 Triangle read by rows, T(n,k) = Sum_{j=k..n} A269940(n,j)*A269939(j,k), for n>=0 and 0<=k<=n.

1, 0, 1, 0, 5, 9, 0, 41, 210, 225, 0, 469, 5115, 14175, 11025, 0, 6889, 142492, 763350, 1455300, 893025, 0, 123605, 4566149, 41943090, 146522250, 212837625, 108056025, 0, 2620169, 166939742, 2462128095, 13973628900, 35936814375, 42141849750, 18261468225

Offset: 0

Peter Luschny, Mar 27 2016

			Triangle starts:
[1]
[0, 1]
[0, 5, 9]
[0, 41, 210, 225]
[0, 469, 5115, 14175, 11025]
[0, 6889, 142492, 763350, 1455300, 893025]

Cf. A001818, A032188, A269939, A269940.

F = lambda n,k,f: sum((-1)^(m+k)*binomial(n+k,n+m)*f(n+m,m) for m in (0..k))
T = lambda n,k: sum(F(n, j, stirling_number1)*F(j, k, stirling_number2) for j in (k..n))
for n in (0..6): print([T(n, k) for k in (0..n)])

T(n,n) = A001818(n).

T(n,1) = A032188(n+1) for n>=1.

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.

Original entry on oeis.org

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)

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.

A118788 Triangle where T(n,k) = n!/(n-k)![x^k] ( x/(2x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 1, 6, 23, 41, 1, 10, 65, 255, 469, 1, 15, 145, 930, 3679, 6889, 1, 21, 280, 2590, 16429, 65247, 123605, 1, 28, 490, 6090, 54789, 344694, 1371887, 2620169, 1, 36, 798, 12726, 151599, 1338330, 8367785, 33347535, 64074901, 1, 45, 1230, 24360

Offset: 0

Paul D. Hanna, Apr 29 2006

Row sums are A118789, where Sum_{n>=0} A118789(n)*x^n/n! = exp( Sum_{n>=1} A032188(n)*x^n/n! ).
Main diagonal is A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.
Secondary diagonal is A118790.

			Row sums e.g.f. equals the exponential of the diagonal e.g.f.:
1 + x + 2*x^2/2! + 9*x^3/3! + 71*x^4/4! +...+ A118789(n)*x^n/n! +...
= exp(x + x^2/2! + 5*x^3/3! + 41*x^4/4! +...+ A032188(n)*x^n/n! +...).
Triangle begins:
  1;
  1, 1;
  1, 3, 5;
  1, 6, 23, 41;
  1, 10, 65, 255, 469;
  1, 15, 145, 930, 3679, 6889;
  1, 21, 280, 2590, 16429, 65247, 123605;
  1, 28, 490, 6090, 54789, 344694, 1371887, 2620169;
  1, 36, 798, 12726, 151599, 1338330, 8367785, 33347535, 64074901;
  ...
Triangle is formed from powers of F(x) = x/(2*x + log(1-x)):
  F(x)^1 = (1) + 1/2*x + 7/12*x^2 + 17/24*x^3 + 629/720*x^4 +...
  F(x)^2 = (1 + x) + 17/12*x^2 + 2*x^3 + 671/240*x^4 +...
  F(x)^3 = (1 + 3/2*x + 5/2*x^2) + 4*x^3 + 1489/240*x^4 +...
  F(x)^4 = (1 + 6/3*x + 23/6*x^2 + 41/6*x^3) + 8351/720*x^4 +...
  F(x)^5 = (1 + 10/4*x + 65/12*x^2 + 255/24*x^3 + 469/24*x^4) +...

Cf. A118789, A118790, A032188.

Third column is A241765.

{T(n,k)=local(x=X+X^2*O(X^(k+2)));n!/(n-k)!*polcoeff((x/(2*x+log(1-x)))^(n+1),k,X)}

Main diagonal has e.g.f.: series_reversion[2*x+log(1-x)].

Conjecture: T(n,k) = Sum_{j=0..k} binomial(n+j, n-k)*A269940(k, j) for 0 <= k <= n. - Mikhail Kurkov, Feb 17 2025

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.

A358622 Regular triangle read by rows. T(n, k) = [[n, k]], where [[n, k]] are the second order Stirling cycle numbers (or second order reciprocal Stirling numbers). T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 0, 24, 20, 0, 0, 0, 0, 120, 130, 15, 0, 0, 0, 0, 720, 924, 210, 0, 0, 0, 0, 0, 5040, 7308, 2380, 105, 0, 0, 0, 0, 0, 40320, 64224, 26432, 2520, 0, 0, 0, 0, 0, 0, 362880, 623376, 303660, 44100, 945, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Nov 23 2022

[[n, k]] are the number of permutations of an n-set having at least two elements in each orbit. These permutations have no fixed points and therefore [[n, k]] is the number of k-orbit derangements of an n-set. This is the definition and notation (doubling the stacked delimiters of the Stirling cycle numbers) as given by Fekete (see link).
The formal definition expresses the second order Stirling cycle numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of first kind' used elsewhere should be dropped in favor of the more systematic one used here.
Also the Bell transform of the factorial numbers with 0! = 0. For the definition of the Bell transform see A264428.

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     0;
[2] 0,     1,     0;
[3] 0,     2,     0,     0;
[4] 0,     6,     3,     0,    0;
[5] 0,    24,    20,     0,    0,  0;
[6] 0,   120,   130,    15,    0,  0,  0;
[7] 0,   720,   924,   210,    0,  0,  0,  0;
[8] 0,  5040,  7308,  2380,  105,  0,  0,  0,  0;
[9] 0, 40320, 64224, 26432, 2520,  0,  0,  0,  0,  0;

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994, thirty-fourth printing 2022.

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. 8.
Antal E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

A008306 is an irregular subtriangle with more information.
Cf. A000166 (row sums), A024000 (alternating row sums).
Cf. A130534, A201637, A341101, A264428, A269940.

P := (n, x) -> (-x)^n*hypergeom([-n, x], [], 1/x):
row := n -> seq(coeff(simplify(P(n, x)), x, k), k = 0..n):
for n from 0 to 9 do row(n) od;
# Alternative:
T := (n, k) -> add(binomial(n, k - j)*abs(Stirling1(n - k + j, j))*(-1)^(k - j), j =  0..k): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Using the e.g.f.:
egf := (exp(t)*(1 - t))^(-z): ser := series(egf, t, 12):
seq(print(seq(n!*coeff(coeff(ser, t, n), z, k), k=0..n)), n = 0..9);
# Using second order Eulerian numbers:
A358622 := proc(n, k) local j;
add(binomial(j, n - 2*k)*combinat:-eulerian2(n - k, j), j = 0..n-k) end:
seq(seq(A358622(n, k), k = 0..n), n = 0..12);
# Using generalized Laguerre polynomials:
P := (n, x) -> (-1)^n*n!*LaguerreL(n, -n - x, -x):
row := n -> seq(coeff(simplify(P(n, x)), x, k), k = 0..n):
seq(print(row(n)), n = 0..9);

# recursion over rows
from functools import cache
@cache
def StirlingCycleOrd2(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 0]
    rov: list[int] = StirlingCycleOrd2(n - 2)
    row: list[int] = StirlingCycleOrd2(n - 1) + [0]
    for k in range(1, n // 2 + 1):
        row[k] = (n - 1) * (rov[k - 1] + row[k])
    return row
for n in range(9): print(StirlingCycleOrd2(n))
# Alternative, using function BellMatrix from A264428.
from math import factorial
def f(k: int) -> int:
    return factorial(k) if k > 0 else 0
print(BellMatrix(f, 9))

T(n, k) = Sum_{j=0..n-k} binomial(j, n - 2*k)*<>, where <> denote the second order Eulerian numbers (extending Knuth's notation).
T(n, k) = [x^n] (-x)^n * hypergeom([-n, x], [], -1/x).
T(n, k) = n!*[z^k][t^n] (exp(t)*(1 - t))^(-z). (Compare with (exp(t)/(1 - t))^z, which is the e.g.f. of the Sylvester polynomials A341101.)
T(n, k) = [x^k] (-1)^n * n! * L(n, -x - n, -x), where L(n, a, x) is the n-th generalized Laguerre polynomial.
T(n, k) = Sum_{j=0..k} binomial(n, k - j)*[n - k + j, j]*(-1)^(k - j), where [n, k] denotes the (signless) Stirling cycle numbers.
T(n, k) = (n - 1) * (T(n-2, k-1) + T(n-1, k)) with suitable boundary conditions.
T(n + k, k) = A269940(n, k), which might be called the Ward cycle numbers.

cp's OEIS Frontend

A269957 Triangle read by rows, T(n,k) = Sum_{j=k..n} A269940(n,j)*A269939(j,k), for n>=0 and 0<=k<=n.

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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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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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A118788 Triangle where T(n,k) = n!/(n-k)!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.

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A268440 Triangle read by rows, T(n,k) = C(2*n,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.

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A358622 Regular triangle read by rows. T(n, k) = [[n, k]], where [[n, k]] are the second order Stirling cycle numbers (or second order reciprocal Stirling numbers). T(n, k) for 0 <= k <= n.

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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.

A118788 Triangle where T(n,k) = n!/(n-k)![x^k] ( x/(2x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.

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.