A002115 -id:A002115 - cp's OEIS frontend

A260877 Square array read by ascending antidiagonals: number of m-shape Euler numbers.

1, 1, -1, 1, -1, 1, 1, -1, 1, -5, 1, -1, 5, -1, 21, 1, -1, 19, -61, 1, -105, 1, -1, 69, -1513, 1385, -1, 635, 1, -1, 251, -33661, 315523, -50521, 1, -4507, 1, -1, 923, -750751, 60376809, -136085041, 2702765, -1, 36457, 1, -1, 3431, -17116009, 11593285251

Offset: 1

Peter Luschny, Aug 09 2015

A set partition of m-shape is a partition of a set with cardinality m*n for some n >= 0 such that the sizes of the blocks are m times the parts of the integer partitions of n. It is ordered if the positions of the blocks are taken into account.
M-shape Euler numbers count the ordered m-shape set partitions which have even length minus the number of such partitions which have odd length.
If m=0 all possible sizes are zero. Thus m-shape Euler numbers count the ordered integer partitions of n into an even number of parts minus the number of ordered integer partitions of n into an odd number of parts (A260845).
If m=1 the set is {1,2,...,n} and the set of all possible sizes are the integer partitions of n. Thus the Euler numbers count the ordered set partitions which have even length minus the set partitions which have odd length (A033999).
If m=2 the set is {1,2,...,2n} and the 2-shape Euler numbers count the ordered set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A028296).

			[ n ] [0   1   2       3         4              5                 6]
[ m ] --------------------------------------------------------------
[ 0 ] [1, -1,  1,     -5,       21,          -105,              635] A260845
[ 1 ] [1, -1,  1,     -1,        1,            -1,                1] A033999
[ 2 ] [1, -1,  5,    -61,     1385,        -50521,          2702765] A028296
[ 3 ] [1, -1, 19,  -1513,   315523,    -136085041,     105261234643] A002115
[ 4 ] [1, -1, 69, -33661, 60376809, -288294050521, 3019098162602349] A211212
         A030662,A211213,  A181991,
For example the number of ordered set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 168, 1680 respectively. Thus A(3,3) = -1 + 168 - 1680 = -1513.
Formatted as a triangle:
[1]
[1, -1]
[1, -1,  1]
[1, -1,  1,    -5]
[1, -1,  5,    -1,   21]
[1, -1, 19,   -61,    1, -105]
[1, -1, 69, -1513, 1385,   -1, 635]

Cf. A002115, A028296, A030662, A033999, A181991, A211212, A211213, A260845, A260833, A260875, A260876.

def A260877(m,n):
    shapes = ([x*m for x in p] for p in Partitions(n).list())
    return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s). cardinality() for s in shapes)
for m in (0..5): print([A260877(m,n) for n in (0..7)])

A318147 Coefficients of the Omega polynomials of order 3, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -9, 10, 0, 477, -756, 280, 0, -74601, 142362, -83160, 15400, 0, 25740261, -55429920, 40900860, -12612600, 1401400, 0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400, 190590400

Offset: 0

Peter Luschny, Aug 22 2018

The name 'Omega polynomial' is not a standard name.

			[0] [1]
[1] [0,            1]
[2] [0,           -9,          10]
[3] [0,          477,        -756,          280]
[4] [0,       -74601,      142362,       -83160,       15400]
[5] [0,     25740261,   -55429920,     40900860,   -12612600,     1401400]
[6] [0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400,190590400]

All row sums are 1, alternating row sums (taken absolute) are A002115.
T(n,1) ~ A293951(n), T(n,n) = A025035(n).
A023531 (m=1), A318146 (m=2), this seq (m=3), A318148 (m=4).

# See A318146 for the missing functions.
FL([seq(CL(OmegaPolynomial(3, n)), n=0..8)]);

(* OmegaPolynomials are defined in A318146 *)
Table[CoefficientList[OmegaPolynomial[3, n], x], {n, 0, 6}] // Flatten

# See A318146 for the function OmegaPolynomial.
[list(OmegaPolynomial(3, n)) for n in (0..6)]

Omega(m, n, z) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m). We consider here the case m = 3 (for other cases see the cross-references).

A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 19, -39, 20, -1513, 4705, -4872, 1680, 315523, -1314807, 2052644, -1422960, 369600, -136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000, 105261234643, -661231439271, 1729495989332, -2410936679424, 1889230062720, -789044256000, 137225088000

Offset: 0

Peter Luschny, Sep 06 2018

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See A318259 and the cross-references for some other members.

			[0] [         1]
[1] [        -1,         1]
[2] [        19,       -39,          20]
[3] [     -1513,      4705,       -4872,       1680]
[4] [    315523,  -1314807,     2052644,   -1422960,     369600]
[5] [-136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000]

Cf. T(n,0) ~ A002115(n) (signed), T(n,n) = A014606.

Cf. A167374 (m=0), A028246 & A163626 (m=1), A318259 (m=2), this seq (m=3).

# uses[EW from A318259]
def A318260row(n): return EW(3, n)
print(flatten([A318260row(n) for n in (0..6)]))

Let P(m,n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x with P(m,0) = 1

and S(n,k) = [x^k]P(3,n), then T(n,k) = Sum_{j=0..k}((-1)^(k-j)*binomial(n-j, n-k)* Sum_{i=0..n}((-1)^i*S(n,i)*binomial(n-i,j))).

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A260877 Square array read by ascending antidiagonals: number of m-shape Euler numbers.

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A318147 Coefficients of the Omega polynomials of order 3, triangle T(n,k) read by rows with 0<=k<=n.

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A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.

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