A278073 -id:A278073 - cp's OEIS frontend

A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

1, 1, 0, 19, 1, 0, 1513, 166, 1, 0, 315523, 52715, 1361, 1, 0, 136085041, 30543236, 1528806, 10916, 1, 0, 105261234643, 29664031413, 2257312622, 42421946, 87375, 1, 0, 132705221399353, 45011574747714, 4637635381695, 153778143100, 1156669095, 699042, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

See the comments in A292604.

			Triangle starts:
[n\k][       0         1         2       3  4  5]
--------------------------------------------------
[0][         1]
[1][         1,        0]
[2][        19,        1,        0]
[3][      1513,      166,        1,     0]
[4][    315523,    52715,     1361,     1,  0]
[5][ 136085041, 30543236,  1528806, 10916,  1, 0]

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} is this triangle, F_{4} = A292606.
First column: A002115. Row sums: A014606. Alternating row sums: A292609.
Cf. A181985, A278073.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f),x):
A292605_row := proc(n) if n = 0 then return [1] fi;
add(A278073(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292605_row(n) od;

# uses[A278073_row from A278073]
def A292605_row(n):
    if n == 0: return [1]
    L = A278073_row(n)
    S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..5): print(A292605_row(n))

F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) for n>0 and F_{3; 0}(x) = 1.

A326477 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 4, 3, 0, 46, 60, 15, 0, 1114, 1848, 840, 105, 0, 46246, 88770, 54180, 12600, 945, 0, 2933074, 6235548, 4574130, 1469160, 207900, 10395, 0, 263817646, 605964450, 505915410, 199849650, 39729690, 3783780, 135135

Offset: 0

Peter Luschny, Jul 08 2019

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 4, 3]
[3] [0, 46, 60, 15]
[4] [0, 1114, 1848, 840, 105]
[5] [0, 46246, 88770, 54180, 12600, 945]
[6] [0, 2933074, 6235548, 4574130, 1469160, 207900, 10395]

Row sums A094088. Alternating row sums A153881 starting at 0.
Main diagonal A001147. Associated set partitions A241171.
A129062 (m=1, associated with A131689), this sequence (m=2), A326587 (m=3, associated with A278073), A326585 (m=4, associated with A278074).

CL := f -> PolynomialTools:-CoefficientList(f, x):
FL := s -> ListTools:-Flatten(s, 1):
StirPochConv := proc(m, n) local P, L; P := proc(m, n) option remember;
`if`(n = 0, 1, add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n)) end:
L := CL(P(m, n)); CL(expand(add(L[k+1]*pochhammer(x,k)/k!, k=0..n))) end:
FL([seq(StirPochConv(2,n), n = 0..7)]);

P[, 0] = 1; P[m, n_] := P[m, n] = Sum[Binomial[m*n, m*k]*P[m, n-k]*x, {k, 1, n}] // Expand;
T[m_][n_] := CoefficientList[P[m, n], x].Table[Pochhammer[x, k]/k!, {k, 0, n}] // CoefficientList[#, x]&;
Table[T[2][n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)

def StirPochConv(m, n):
    z = var('z'); R = ZZ[x]
    F = [i/m for i in (1..m-1)]
    H = hypergeometric([], F, (z/m)^m)
    P = R(factorial(m*n)*taylor(exp(x*(H-1)), z, 0, m*n + 1).coefficient(z, m*n))
    L = P.list()
    S = sum(L[k]*rising_factorial(x,k) for k in (0..n))
    return expand(S).list()
for n in (0..6): print(StirPochConv(2, n))

For m >= 1 let P(m,0) = 1 and P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x for n > 0. Then T_{m}(n, k) = Sum_{k=0..n} ([x^k]P(m, n))*rf(x,k)/k! where rf(x,k) are the rising factorial powers. T(n, k) = T_{2}(n, k).

A326585 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 4 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 36, 35, 0, 12046, 17820, 5775, 0, 16674906, 30263480, 16216200, 2627625, 0, 65544211366, 135417565890, 93516348900, 26189163000, 2546168625, 0, 588586227465426, 1334168329550300, 1083314031995250, 402794176785000, 69571511509500, 4509264634875

Offset: 0

Peter Luschny, Jul 21 2019

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 36, 35]
[3] [0, 12046, 17820, 5775]
[4] [0, 16674906, 30263480, 16216200, 2627625]
[5] [0, 65544211366, 135417565890, 93516348900, 26189163000, 2546168625]
[6] [0, 588586227465426, 1334168329550300, 1083314031995250, 402794176785000, 69571511509500, 4509264634875]

Row sums A243665. Main diagonal A025036.

A129062 (m=1, associated with A131689), A326477 (m=2, associated with A241171), A326587 (m=3, associated with A278073), this sequence (m=4, associated with A278074).

Maple
```
# See A326477.
```

T(n, k) = T_{4}(n, k) where T_{m}(n, k) is defined in A326477.

A326587 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 3 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 11, 10, 0, 645, 924, 280, 0, 111563, 197802, 101640, 15400, 0, 42567981, 86271640, 57717660, 15415400, 1401400, 0, 30342678923, 67630651098, 53492240256, 19158419280, 3144741600, 190590400

Offset: 0

Peter Luschny, Jul 20 2019

			Triangle starts:
0 [1]
1 [0, 1]
2 [0, 11, 10]
3 [0, 645, 924, 280]
4 [0, 111563, 197802, 101640, 15400]
5 [0, 42567981, 86271640, 57717660, 15415400, 1401400]
6 [0, 30342678923, 67630651098, 53492240256, 19158419280, 3144741600, 190590400]

Row sums A243664. Main diagonal A025035.

A129062 (m=1, associated with A131689), A326477 (m=2, associated with A241171), this sequence (m=3, associated with A278073), A326585 (m=4, associated with A278074).

Maple
```
# See A326477.
```

T(n, k) = T_{3}(n, k) where T_{m}(n, k) is defined in A326477.

A327023 Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

Original entry on oeis.org

1, 1, 1, 20, 1, 168, 1680, 1, 440, 924, 55440, 369600, 1, 910, 10010, 300300, 1261260, 33633600, 168168000, 1, 1632, 37128, 48620, 1113840, 24504480, 17153136, 326726400, 2058376320, 34306272000, 137225088000

Offset: 0

Peter Luschny, Aug 27 2019

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 3. For instance 3*P(4, .) = [[12], [9, 3], [6, 6], [6, 3, 3], [3, 3, 3, 3]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;   20]
[3] [1;  168;  1680]
[4] [1;  440,   924;  55440;  369600]
[5] [1;  910, 10010; 300300, 1261260; 33633600; 168168000]
[6] [1; 1632, 37128,  48620; 1113840, 24504480,  17153136; 326726400, 2058376320;
     34306272000; 137225088000]
.
T(4, 1) = 440 because [9, 3] is the integer partition 3*P(4, 1) in the canonical order and there are 220 set partitions which have the shape [9, 3]. Finally, since the order of the sets is taken into account, one gets 2!*220 = 440.

Row sums: A243664, alternating row sums: A002115, main diagonal: A014606, central column A281479, by length: A278073.
Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), this sequence (m=3), A327024 (m=4).
Cf. A080577, A000041, A072233.

# uses[GenOrdSetPart from A327022]
def A327023row(n): return GenOrdSetPart(3, n)
for n in (0..6): print(A327023row(n))

A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).

Original entry on oeis.org

1, 1, 1, 3, 6, 3, 13, 39, 39, 13, 75, 300, 450, 300, 75, 541, 2705, 5410, 5410, 2705, 541, 4683, 28098, 70245, 93660, 70245, 28098, 4683, 47293, 331051, 993153, 1655255, 1655255, 993153, 331051, 47293, 545835, 4366680, 15283380, 30566760, 38208450, 30566760, 15283380, 4366680, 545835

Offset: 0

Peter Luschny, Apr 26 2023

			[0]    1;
[1]    1,     1;
[2]    3,     6,     3;
[3]   13,    39,    39,    13;
[4]   75,   300,   450,   300,    75;
[5]  541,  2705,  5410,  5410,  2705,   541;
[6] 4683, 28098, 70245, 93660, 70245, 28098, 4683;

Family of triangles: A055372 (m=0, Pascal), this sequence (m=1, Fubini), A362586 (m=2, Joffe), A362849 (m=3, A278073).

Cf. A000670 (column 0 and main diagonal), A216794 (row sums).

def TransOrdPart(m, n) -> list[int]:
    @cached_function
    def P(m: int, n: int):
        R = PolynomialRing(ZZ, "x")
        if n == 0: return R(1)
        return R(sum(binomial(m * n, m * k) * P(m, n - k) * x
                 for k in range(1, n + 1)))
    T = P(m, n)
    def C(k) -> int:
        return sum(T[j] * binomial(n, k) for j in range(n + 1))
    return [C(k) for k in range(n+1)]
def A362585(n) -> list[int]: return TransOrdPart(1, n)
for n in range(6): print(A362585(n))

A362849 Triangle read by rows, T(n, k) = A243664(n) * binomial(n, k).

Original entry on oeis.org

1, 1, 1, 21, 42, 21, 1849, 5547, 5547, 1849, 426405, 1705620, 2558430, 1705620, 426405, 203374081, 1016870405, 2033740810, 2033740810, 1016870405, 203374081, 173959321557, 1043755929342, 2609389823355, 3479186431140, 2609389823355, 1043755929342, 173959321557

Offset: 0

Peter Luschny, May 05 2023

			[0]         1;
[1]         1,          1;
[2]        21,         42,         21;
[3]      1849,       5547,       5547,       1849;
[4]    426405,    1705620,    2558430,    1705620,     426405;
[5] 203374081, 1016870405, 2033740810, 2033740810, 1016870405, 203374081;

Family of triangles: A055372 (m=0, Pascal), A362585 (m=1, Fubini), A362586 (m=2, Joffe), this sequence (m=3, A278073).

Cf. A243664 (column 0 and main diagonal).

# uses[TransOrdPart from A362585]
def A362849(n) -> list[int]: return TransOrdPart(3, n)
for n in range(6): print(A362849(n))

A326717 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 5 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 127, 126, 0, 255256, 381381, 126126, 0, 2979852651, 5447453786, 2956465512, 488864376, 0, 127156445503275, 264284637872750, 184292523727620, 52359004217520, 5194672859376

Offset: 0

Peter Luschny, Jul 21 2019

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 127, 126]
[3] [0, 255256, 381381, 126126]
[4] [0, 2979852651, 5447453786, 2956465512, 488864376]
[5] [0, 127156445503275, 264284637872750, 184292523727620, 52359004217520, 5194672859376]
[6] [0, 15160169962750251082, 34544220081315967665, 28276496764200664980, 10634436034307385300, 1865368063755476280, 123378675083039376]

Row sums A243666. Main diagonal A025037.

A129062 (m=1, associated with A131689), A326477 (m=2, associated with A241171), A326587 (m=3, associated with A278073), A326585 (m=4, associated with A278074), this sequence (m=5).

Maple
```
# See A326477.
```

T(n, k) = T_{5}(n, k) where T_{m}(n, k) is defined in A326477.

cp's OEIS Frontend

A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

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A326477 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.

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A326585 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 4 and 0 <= k <= n.

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A326587 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 3 and 0 <= k <= n.

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A327023 Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

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A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).

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SageMath

A362849 Triangle read by rows, T(n, k) = A243664(n) * binomial(n, k).

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A326717 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 5 and 0 <= k <= n.

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