A024000 -id:A024000 - cp's OEIS frontend

A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 2, 1, -1, 1, 5, 1, -2, 2, 1, 15, 1, -3, 5, -3, 4, 52, 1, -4, 10, -13, 7, 11, 203, 1, -5, 17, -35, 36, -10, 41, 877, 1, -6, 26, -75, 127, -101, 31, 162, 4140, 1, -7, 37, -139, 340, -472, 293, -21, 715, 21147, 1, -8, 50, -233, 759, -1573, 1787, -848, 204, 3425, 115975

Offset: 0

Alois P. Heinz, Mar 23 2023

			Square array A(n,k) begins:
    1,   1,   1,    1,     1,      1,       1,       1, ...
    1,   0,  -1,   -2,    -3,     -4,      -5,      -6, ...
    2,   1,   2,    5,    10,     17,      26,      37, ...
    5,   1,  -3,  -13,   -35,    -75,    -139,    -233, ...
   15,   4,   7,   36,   127,    340,     759,    1492, ...
   52,  11, -10, -101,  -472,  -1573,   -4214,   -9685, ...
  203,  41,  31,  293,  1787,   7393,   23711,   63581, ...
  877, 162, -21, -848, -6855, -35178, -134873, -421356, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns k=0-5 give: A000110, A000296, A126617, A346738, A346739, A346740.
Rows n=0-2 give: A000012, A024000, A160457.
Main diagonal gives A290219.
Antidiagonal sums give A361380.
Cf. A108087.

T:= func< n,k | (&+[(-k)^j*Binomial(n,j)*Bell(n-j): j in [0..n]]) >;
A361781:= func< n,k | T(k, n-k) >;
[A361781(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024

A:= proc(n, k) option remember; uses combinat;
      add(binomial(n, j)*(-k)^j*bell(n-j), j=0..n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
A:= (n, k)-> b(n, -k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

T[n_, k_]:= T[n, k]= If[k==0, BellB[n], Sum[(-k)^j*Binomial[n,j]*BellB[n-j], {j,0,n}]];
A361781[n_, k_]= T[k, n-k];
Table[A361781[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 12 2024 *)

def T(n,k): return sum( (-k)^j*binomial(n,j)*bell_number(n-j) for j in range(n+1))
def A361781(n, k): return T(k, n-k)
flatten([[A361781(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

E.g.f. of column k: exp(exp(x) - k*x - 1).

A(n,k) = Sum_{j=0..n} (-k)^j*binomial(n,j)*Bell(n-j).

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.

A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -4, 1, 1, 1, -3, -8, -7, -4, 1, 1, 1, -4, -11, -8, 1, 0, 1, 1, 1, -5, -14, -7, 16, 23, 8, 1, 1, 1, -6, -17, -4, 41, 64, 43, 16, 1, 1, 1, -7, -20, 1, 76, 117, 64, 17, 16, 1, 1, 1, -8, -23, 8, 121, 176, 29, -128, -95, 0, 1

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com) and Robert G. Wilson v, Jan 02 2013

.j\k.........0..1...2....3...4....5....6......7.......8......9......10
.0: A000012..1..1...1....1...1....1....1......1.......1......1.......1
-1: A146559..1..1...0...-2..-4...-4....0......8......16.....16.......0
-2: A087455..1..1..-1...-5..-7....1...23.....43......17....-95....-241
-3: A138230..1..1..-2...-8..-8...16...64.....64....-128...-512....-512
-4: A006495..1..1..-3..-11..-7...41..117.....29....-527..-1199.....237
-5: A138229..1..1..-4..-14..-4...76..176...-104...-1264..-1904....3776
-6: A090592..1..1..-5..-17...1..121..235...-377...-2399..-2159...12475
-7: A090590..1..1..-6..-20...8..176..288...-832...-3968..-1280...29184
-8: A025172..1..1..-7..-23..17..241..329..-1511...-5983...1633...57113
-9: A120743..1..1..-8..-26..28..316..352..-2456...-8432...7696...99712
-10: ........1..1..-9..-29..41..401..351..-3709..-11279..18241..160551

Cf. A084097,
Rows: A000012, A146559, A087455, A138230, A006495, A138229, A090592, A090590, A025172, A120743.
Columns: A000012, A000012, A024000, 1-3n, A028884, A134593.

T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; Table[ T[ -j + k, k], {j, 0, 11}, {k, 0, j}] // Flatten

A373164 Triangle read by rows: the exponential almost-Riordan array ( 1 | 2 - exp(x), x ).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, -1, -3, -3, 1, 0, -1, -4, -6, -4, 1, 0, -1, -5, -10, -10, -5, 1, 0, -1, -6, -15, -20, -15, -6, 1, 0, -1, -7, -21, -35, -35, -21, -7, 1, 0, -1, -8, -28, -56, -70, -56, -28, -8, 1, 0, -1, -9, -36, -84, -126, -126, -84, -36, -9, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
  1;
  0,  1;
  0, -1,  1;
  0, -1, -2,   1;
  0, -1, -3,  -3,   1;
  0, -1, -4,  -6,  -4,   1;
  0, -1, -5, -10, -10,  -5,   1;
  0, -1, -6, -15, -20, -15,  -6,  1;
  0, -1, -7, -21, -35, -35, -21, -7, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 8.

Cf. A000012 (right diagonal), A024000 (subdiagonal), A122958 (row sums), A153881 (k=1).

Triangle A154926 with 1st column A000007.

T[n_,0]:=KroneckerDelta[n,0]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[(2-Exp[x])x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

T(n,0) = A000007(n); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] (2-exp(x))*x^(k-1).

cp's OEIS Frontend

A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

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A373164 Triangle read by rows: the exponential almost-Riordan array ( 1 | 2 - exp(x), x ).

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