A156289 -id:A156289 - cp's OEIS frontend

A257490 Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.

1, 1, 3, 1, 15, 15, 1, 28, 35, 210, 105, 1, 45, 210, 630, 1575, 3150, 945, 1, 66, 495, 462, 1485, 13860, 5775, 13860, 51975, 51975, 10395, 1, 91, 1001, 3003, 3003, 45045, 42042, 105105, 45045, 630630, 525525, 315315, 1576575, 945945, 135135

Offset: 1

Hartmut F. W. Hoft, Apr 26 2015

The length of row n is given by A000041(n).
Each entry in this irregular triangle is the quotient of the respective entries in A257468 and A096162, which is the multinomial called M_3 in Abramowitz-Stegun.
Has the same structure as the triangles in A036036, A096162, A115621 and A257468.

			Brackets group all partitions of the same length when there is more than one partition.
n/m  1    2          3           4    5
1:   1
2:   1    3
3:   1   15         15
4:   1  [28  35]   210         105
5:   1  [45 210]  [630 1575]  3150  945
...
n = 6:  1 [66 495 462] [1485 13860 5775] [13860 51975] 51975  0395
Replacing the bracketed numbers by their sums yields the triangle of A156289.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], pp. 831-832.

Cf. A000041, A036040, A036036, A096162, A115621, A156289, A257468.

(* triangle2574868[] and triangle096162[] are defined as functions triangle[] in the respective sequences A257468 and A096162 *)
triangle[n_] := triangle257468[n]/triangle096162[n]
a[n_] := Flatten[triangle[n]]
a[7] (* data *)

Edited by Wolfdieter Lang, May 11 2015

A327417 a(n) = A291451(2*n, n).

Original entry on oeis.org

1, 1, 682, 7128576, 429120851544, 94066556834970720, 57496301859366489159040, 82247725949165261902606309120, 243263294602173417290925789755652480, 1356449073308047884259226117174893156252800, 13275987570857688650109290727617026478737341900800

Offset: 0

Peter Luschny, Sep 14 2019

nonn

Cf. A007820 (m=1), A327416 (m=2), this sequence (m=3), A327418 (m=4).

Associated triangles: A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).

# uses[P from A327416]
def A327417(n): return P(3, 2*n).list()[n]//factorial(n)
print([A327417(n) for n in range(11)])

A327418 a(n) = A291452(2*n, n).

Original entry on oeis.org

1, 1, 8255, 2941884000, 11957867341948125, 294040106448733743008625, 30188472144950452369737153667500, 10143939867539251013312279527292897925000, 9389957475743686923255643914812959599614184703125, 21058194888200109612591474039339954750056969537259132421875

Offset: 0

Peter Luschny, Sep 14 2019

nonn

Cf. A007820 (m=1), A327416 (m=2), A327417 (m=3), this sequence (m=4).

Associated triangles: A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).

# uses[P from A327416]
def A327418(n): return P(4, 2*n).list()[n]//factorial(n)
print([A327418(n) for n in range(10)])

A361948 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} binomial((j + 1)*n - 1, n - 1) if n >= 1, and A(0, k) = 1 for all k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 10, 15, 1, 1, 1, 1, 35, 280, 105, 1, 1, 1, 1, 126, 5775, 15400, 945, 1, 1, 1, 1, 462, 126126, 2627625, 1401400, 10395, 1, 1, 1, 1, 1716, 2858856, 488864376, 2546168625, 190590400, 135135, 1, 1

Offset: 0

Peter Luschny, Apr 13 2023

Row n gives the leading coefficients of the set partition polynomials of type n. The sequence of these polynomial sequences starts: A097805, A048993, A156289, A291451, A291452, ...

			Array A(n, k) starts:
  [0] 1, 1,   1,       1,           1,                 1, ...
  [1] 1, 1,   1,       1,           1,                 1, ...
  [2] 1, 1,   3,      15,         105,               945, ...  A001147
  [3] 1, 1,  10,     280,       15400,           1401400, ...  A025035
  [4] 1, 1,  35,    5775,     2627625,        2546168625, ...  A025036
  [5] 1, 1, 126,  126126,   488864376,     5194672859376, ...  A025037
  [6] 1, 1, 462, 2858856, 96197645544, 11423951396577720, ...  A025038
.
Triangle A(n-k, k) starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1,  1;
  [3] 1, 1,  1,   1;
  [4] 1, 1,  3,   1,   1;
  [5] 1, 1, 10,  15,   1, 1;
  [6] 1, 1, 35, 280, 105, 1, 1;

Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.
Antoine Chambert-Loir, Combinatorics of partitions, blog post 2024.
Alexander Karpov, Generalized knockout tournament seedings, International Journal of Computer Science in Sport, vol. 17(2), 2018.

Cf. A060540 (subarray), A370407 (antidiagonal sums, row sums).
Cf. A001147 (row 2), A025035 (row 3), A025036 (row 4), A025037 (row 5), A025038 (row 6), A025039 (row 7), A025040 (row 8), A025041 (row 9).
Cf. A088218 (column 2), A060542 (column 3), A082368 (column 4), A322252 (column 5), A057599 (main diagonal).
Cf. A097805, A048993, A156289, A291451, A291452.

A := (n, k) -> mul(binomial((j + 1)*n - 1, n - 1), j = 0..k-1):
seq(seq(A(n-k, k), k = 0..n), n = 0..9);
# Alternative, using recursion:
A := proc(n, k) local P; P := proc(n, k) option remember;
if n = 0 then return x^k*k! fi; if k = 0 then 1 else add(binomial(n*k, n*j)*
P(n,k-j)*x, j=1..k) fi end: coeff(P(n, k), x, k) / k! end:
seq(print(seq(A(n, k), k = 0..5)), n = 0..6);
# Alternative, using exponential generating function:
egf := n -> ifelse(n=0, 1, exp(x^n/n!)): ser := n -> series(egf(n), x, 8*n):
row := n -> local k; seq((n*k)!*coeff(ser(n), x, n*k), k = 0..6):
for n from 0 to 6 do [n], row(n) od;  # Peter Luschny, Aug 15 2024

A[n_, k_] := Product[Binomial[n (j + 1) - 1, n - 1], {j, 0, k - 1}]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 13 2023 *)

def Arow(n, size):
    if n == 0: return [1] * size
    return [prod(binomial((j + 1)*n - 1, n - 1) for j in range(k)) for k in range(size)]
for n in range(7): print(Arow(n, 7))
# Alternative, using exponential generating function:
def SetPolyLeadCoeff(m, n):
    x, z = var("x, z")
    if m == 0: return 1
    w = exp(2 * pi * I / m)
    o = sum(exp(z * w ** k) for k in range(m)) / m
    t = exp(x * (o - 1)).taylor(z, 0, m*n)
    p = factorial(m*n) * t.coefficient(z, m*n)
    return p.leading_coefficient(x)
for m in range(7):
    print([SetPolyLeadCoeff(m, k) for k in range(6)])

A(n, k) = (1/k!) * [x^k] P(n, k), where P(n, k) = k!*x^k if n = 0 and otherwise 1 if k = 0 and otherwise Sum_{j=1..k} binomial(n*k, n*j)*P(n, k-j)*x.

A(n, k) = (n*k)!*[x^(n*k)] exp(x^n/n!) for n >= 1. - Peter Luschny, Aug 15 2024

A156290 Triangle read by rows: alternating binomial coefficients with signs.

Original entry on oeis.org

1, -4, 1, 15, -6, 1, -56, 28, -8, 1, 210, -120, 45, -10, 1, -792, 495, -220, 66, -12, 1, 3003, -2002, 1001, -364, 91, -14, 1, -11440, 8008, -4368, 1820, -560, 120, -16, 1, 43758, -31824, 18564, -8568, 3060, -816, 153, -18, 1, -167960, 125970, -77520

Offset: 1

Hartmut F. W. Hoft, Feb 07 2009

Alternating binomial coefficients in the closed form expression for sequence A156289.

The Example lines below show the connection with Pascal's triangle A007318.

			R(2,1)=-4, R(3,3)=1, R(4,2)=28.
Here is Pascal's triangle with the entries in the present triangle preceded by a *:
......................1
.....................1, 1
...................1, 2,*1
.................1, 3, 3, 1
................1, 4, 6,*4,*1
..............1, 5, 10, 10, 5, 1
............1, 6, 15, 20,*15,*6,*1
..........1, 7, 21, 35, 35, 21, 7, 1
........1, 8, 28, 56, 70,*56,*28,*8,*1
...

T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.

Coefficient factor in elements of sequence A156289, the inverse of lower triangular matrix A156308.

Cf. A007318.

R[m_] := Flatten[Table[(-1)^(k + j) Binomial[2 k, k + j], {k, 1, m}, {j, 1, k}]]

R(k,j)=(-1)^(k+j)*Binomial(2k,k+j), for 1<= j<=k, and 0 otherwise.

Edited by N. J. A. Sloane, Apr 05 2011

A362370 Triangle read by rows. T(n, k) = ([x^k] P(n, x)) // k! where P(n, x) = Sum_{k=1..n} P(n - k, x) * x if n >= 1 and P(0, x) = 1. The notation 's // t' means integer division and is a shortcut for 'floor(s/t)'.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 1, 4, 4, 2, 0, 0, 0, 0, 0, 0, 1, 4, 6, 3, 1, 0, 0, 0, 0, 0, 0, 1, 5, 7, 5, 1, 0, 0, 0, 0, 0, 0, 0, 1, 5, 9, 6, 2, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Apr 17 2023

Row n gives the coefficients of the set partition polynomials of type m = 0 (the base case). The sequence of these polynomial sequences starts: this sequence, A048993, A156289, A291451, A291452, ...

			Triangle T(n, k) starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 0]
  [3] [0, 1, 1, 0]
  [4] [0, 1, 1, 0, 0]
  [5] [0, 1, 2, 1, 0, 0]
  [6] [0, 1, 2, 1, 0, 0, 0]
  [7] [0, 1, 3, 2, 0, 0, 0, 0]
  [8] [0, 1, 3, 3, 1, 0, 0, 0, 0]
  [9] [0, 1, 4, 4, 2, 0, 0, 0, 0, 0]

Cf. A097805, A362307 (row sums).

Cf. the family of partition polynomials: this sequence (m=0), A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).

T := (n, k) -> iquo(binomial(n - 1, k - 1), k!):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);

R = PowerSeriesRing(ZZ, "x")
x = R.gen().O(33)
@cached_function
def p(n) -> Polynomial:
    if n == 0: return R(1)
    return sum(p(n - k) * x for k in range(1, n + 1))
def A362370_row(n) -> list[int]:
    L = p(n).list()
    return [L[k] // factorial(k) for k in range(n + 1)]
for n in range(10):
    print(A362370_row(n))

T(n, k) = floor(A097805(n, k) / k!).

A318257 Triangle read by rows, expansion of the e.g.f. given below related to partitions of {1,2,...,5n} into sets of size 5, nonzero coefficients of z.

Original entry on oeis.org

1, 0, 1, 0, 1, 126, 0, 1, 3003, 126126, 0, 1, 107882, 23279256, 488864376, 0, 1, 3321890, 5319906900, 412275623760, 5194672859376, 0, 1, 107746281, 1394769716340, 369277150181940, 14687937509885640, 123378675083039376

Offset: 0

Peter Luschny, Aug 22 2018

			[0] [1]
[1] [0, 1]
[2] [0, 1,     126]
[3] [0, 1,    3003,     126126]
[4] [0, 1,  107882,   23279256,    488864376]
[5] [0, 1, 3321890, 5319906900, 412275623760, 5194672859376]

Cf. A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4), this seq (m=5).

CL := p -> PolynomialTools:-CoefficientList(p, x):
FL := p -> ListTools:-Flatten(p):
f := z -> (1/5)*(exp(z)+2*(+exp(1/4*z*(5^(1/2)-1))*cos(1/4*z*2^(1/2)*
(5+5^(1/2))^(1/2))+exp(-1/4*z*(5^(1/2)+1))*cos(1/4*z*2^(1/2)*(5-5^(1/2))^(1/2)))):
gf := exp(x*(f(z)-1)): ser := series(gf, z, 48):
FL([seq(CL(sort(expand((5*n)!*coeff(ser, z, n*5)), [x], ascending)),n=0..7)]);

cp's OEIS Frontend

A257490 Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.

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A327417 a(n) = A291451(2*n, n).

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A327418 a(n) = A291452(2*n, n).

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A361948 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} binomial((j + 1)*n - 1, n - 1) if n >= 1, and A(0, k) = 1 for all k.

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A156290 Triangle read by rows: alternating binomial coefficients with signs.

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A362370 Triangle read by rows. T(n, k) = ([x^k] P(n, x)) // k! where P(n, x) = Sum_{k=1..n} P(n - k, x) * x if n >= 1 and P(0, x) = 1. The notation 's // t' means integer division and is a shortcut for 'floor(s/t)'.

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A318257 Triangle read by rows, expansion of the e.g.f. given below related to partitions of {1,2,...,5n} into sets of size 5, nonzero coefficients of z.

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