A260876 -id:A260876 - cp's OEIS frontend

A326996 Main diagonal of A260876.

1, 1, 4, 365, 3086331, 5612271711927, 3829797188212731601783, 1478967550753025951356611840021161, 452501475882033837823819972299248189399008553836, 146630849220097180917463638217405949960396188742877031073909770851

Offset: 0

Peter Luschny, Aug 13 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..27

Cf. A260876, A327001.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
       binomial(k*n-1, k*j-1)*b(n-j, k), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..10);  # Alois P. Heinz, Aug 14 2019

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Binomial[k n - 1, k j - 1] b[n - j, k], {j, 1, n}]];
a[n_] := b[n, n];
a /@ Range[0, 10] (* Jean-François Alcover, Nov 07 2020, after Maple *)

a(n) = A260876(n,n).

a(n) = (n^2)! * [x^(n^2)] exp(Sum_{k>=1} x^(n*k) / (n*k)!). - Ilya Gutkovskiy, Feb 09 2020

A260878 Number of set partitions of {1, 2, ..., 2*n} with sizes in {[n, n], [2n]}.

Original entry on oeis.org

2, 2, 4, 11, 36, 127, 463, 1717, 6436, 24311, 92379, 352717, 1352079, 5200301, 20058301, 77558761, 300540196, 1166803111, 4537567651, 17672631901, 68923264411, 269128937221, 1052049481861, 4116715363801, 16123801841551, 63205303218877, 247959266474053

Offset: 0

Peter Luschny, Aug 02 2015

Third column in A260876.

			The set partitions counted by a(3) = 11 are: {{1, 2, 3, 4, 5, 6}},
{{1, 2, 4}, {3, 5, 6}}, {{1, 2, 3}, {4, 5, 6}}, {{1, 3, 4}, {2, 5, 6}},
{{1, 3, 5}, {2, 4, 6}}, {{1, 4, 5}, {2, 3, 6}}, {{1, 5, 6}, {2, 3, 4}},
{{1, 4, 6}, {2, 3, 5}}, {{1, 3, 6}, {2, 4, 5}}, {{1, 2, 6}, {3, 4, 5}},
{{1, 2, 5}, {3, 4, 6}}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Hans Salié, Über die Koeffizienten der Blasiusschen Reihe, Math. Nachr. 1955, (14, 4-6), 241--248.

a(n) = A112849(n) for n >= 2. - Alois P. Heinz, Aug 06 2015
a(n) = A052473(n+2) - 1.
a(n) = A088218(n) + 1.
a(n) = (-1)^n*A110556(n) + 1.
a(n+1) - a(n) = A097613(n+1) for n > 0.
Cf. A260876, A001700.
Cf. A323230 (d=0), this sequence (d=1), A323229 (d=2).

a := proc(n) option remember;
if n < 2 then [2, 2][n+1] else ((4*n - 2)*a(n-1) - 3*n + 2)/n fi end:
seq(a(n), n=0..26); # Or:
egf := n -> exp(exp(x)*(1 - (GAMMA(n,x)/GAMMA(n)))):
a := n -> `if`(n<2, 2, (2*n)!*coeff(series(egf(n), x, 2*n+1), x, 2*n)):
seq(a(n), n=0..26); # Peter Luschny, Aug 02 2019

Table[Binomial[2 n - 1, n] + 1, {n, 0, 26}] (* or *)
CoefficientList[Series[(4 x^2 - 13 x + 3 + Sqrt[(1 - 4 x) (x - 1)^2])/(2 (4 x - 1) (x - 1)), {x, 0, 26}], x] (* Michael De Vlieger, Feb 26 2017 *)

Sage
```
print([A260876(n,2) for n in (0..30)])
```

# Alternative:
def A260878():
    a, f, s, n = 2, 2, 1, 1
    yield a
    while True:
        yield a
        f += 4; s += 3; n += 1
        a = (f*a - s)/n
a = A260878()
print([next(a) for n in range(27)]) # Peter Luschny, Aug 02 2019

G.f.: (4*x^2 - 13*x + 3 + sqrt((1 - 4*x)*(x - 1)^2))/(2*(4*x - 1)*(x - 1)). - Alois P. Heinz, Aug 06 2015
a(n) = Binomial(2*n-1, n) + 1. - Vladimir Kruchinin, Feb 26 2017
The generating function G(x) satisfies the differential equation x^3 + 2*x = (4*x^4 - 9*x^3 + 6*x^2 - x)*diff(G(x), x) + (2*x^3 - 4*x^2 + 2*x)*G(x). - Peter Luschny, Feb 12 2019
From Peter Luschny, Aug 02 2019: (Start)
a(n) = ((4*n - 2)*a(n-1) - 3*n + 2)/n for n >= 2.
a(n) = (2*n)! * [x^(2*n)] exp(exp(x)*(1 - (Gamma(n,x)/Gamma(n)))) for n >= 2.
a(n) ~ 4^n/sqrt(4*Pi*n). More precise asymptotic estimates are:
1 + (4^n/sqrt(n*Pi)) * (1/2 - 1/(16*n) * (1 - 1/(16*n))), and
1 + 4^n*(2 - 2/N^2 + 21/N^4 - 671/N^6) / sqrt(2*N*Pi) with N = 8*n + 2.
Let b(n) = binomial(2*(n-1), n-1) + 1 = A323230(n) for n >= 0. Then by Salié:
p divides a(p+k) - b(k+1) if p is a prime > k and 0 <= k <= 4.
Conjecture: p divides a(p+5) - b(6) if p is a prime > b(6).
If p is a prime divisor of n then a(n) == a(n/p) (mod p) (by Salié, theorem 2).
(End)
From Peter Bala, Apr 20 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^k * 3*n/(2*n + k) * binomial(2*n+k, n-k) for n >= 1.
a(n) = Sum_{k = 0..n} (-1)^k * 3*n/(n + 2*k) * binomial(2*n+k-1, n-k) for n >= 1.
(-1)^n * a(n) equals the n-th order Taylor polynomial (centered at 0) of 1/c(x)^(3*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. (End)

A327001 Generalized Bell numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 4, 5, 8, 1, 1, 11, 31, 15, 16, 1, 1, 36, 365, 379, 52, 32, 1, 1, 127, 6271, 25323, 6556, 203, 64, 1, 1, 463, 129130, 3086331, 3068521, 150349, 877, 128, 1, 1, 1717, 2877421, 512251515, 3309362716, 583027547, 4373461, 4140, 256

Offset: 0

Peter Luschny, Aug 12 2019

			[n\k][0  1   2        3        4           5             6]
[ - ] -----------------------------------------------------
[ 0 ] 1, 1,  2,       4,       8,         16,            32  A011782
[ 1 ] 1, 1,  2,       5,      15,         52,           203  A000110
[ 2 ] 1, 1,  4,      31,     379,       6556,        150349  A005046
[ 3 ] 1, 1, 11,     365,   25323,    3068521,     583027547  A291973
[ 4 ] 1, 1, 36,    6271, 3086331, 3309362716, 6626013560301  A291975
       A260878, A326998,
Formatted as a triangle:
[1]
[1, 1]
[1, 1,   2]
[1, 1,   2,    4]
[1, 1,   4,    5,     8]
[1, 1,  11,   31,    15,    16]
[1, 1,  36,  365,   379,   52,  32]
[1, 1, 127, 6271, 25323, 6556, 203, 64]

A260876 (variant based on shapes).
Rows include: A011782, A000110, A005046, A291973, A291975.
Columns include: A260878, A326998.
Cf. A327000.

A327001 := proc(n, k) option remember; if k = 0 then return 1 fi;
add(binomial(n*k - 1, n*j) * A327001(n, j), j = 0..k-1) end:
for n from 0 to 6 do seq(A327001(n, k), k=0..6) od; # row-wise

A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k-1, n*j]*A[n, j], {j, 0, k-1}]];
Table[A[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 27 2022 *)

A(n, k) = Sum_{j=0..k-1} binomial(n*k - 1, n*j) * A(n, j) for k > 0, A(n, 0) = 1.

A260875 Square array read by ascending antidiagonals: number of m-shape complementary Bell numbers.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, 0, -1, 1, -1, 2, 1, 1, 1, -1, 9, -1, 1, -1, 1, -1, 34, -197, -43, -2, 1, 1, -1, 125, -5281, 6841, 254, -9, -1, 1, -1, 461, -123124, 2185429, -254801, 4157, -9, 2, 1, -1, 1715, -2840293, 465693001, -1854147586, -3000807, -70981, 50, -2

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.
M-complementary Bell numbers count the 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 in this case the complementary Bell numbers count the integer partitions of n into an even number of parts minus the number of integer partitions of n into an odd number of parts (A081362).
If m=1 the set is {1,2,...,n} and the complementary Bell numbers count the set partitions which have even length minus the set partitions which have odd length (A000587).
If m=2 the set is {1,2,...,2n} and the complementary Bell numbers count the set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A260884).

			[ n ] [ 0   1   2      3        4            5              6]
[ m ] --------------------------------------------------------
[ 0 ] [ 1, -1,  0,    -1,       1,          -1,             1] A081362
[ 1 ] [ 1, -1,  0,     1,       1,          -2,            -9] A000587
[ 2 ] [ 1, -1,  2,    -1,     -43,         254,          4157] A260884
[ 3 ] [ 1, -1,  9,  -197,    6841,     -254801,      -3000807]
[ 4 ] [ 1, -1, 34, -5281, 2185429, -1854147586, 2755045819549]
      A010763,
For example the number of set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 84, 280 respectively. Thus A(3,3) = -1 + 84 - 280 = -197.
Formatted as a triangle:
[1]
[1, -1]
[1, -1,   0]
[1, -1,   0,    -1]
[1, -1,   2,     1,    1]
[1, -1,   9,    -1,    1,  -1]
[1, -1,  34,  -197,  -43,  -2,  1]
[1, -1, 125, -5281, 6841, 254, -9, -1]

Cf. A000587, A010763, A081362, A260877, A260833, A260884, A260876.

def A260875(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum((-1)^len(s)*SetPartitions(sum(s),s).cardinality() for s in shapes)
for m in (0..4): print([A260875(m,n) for n in (0..6)])

A260883 Number of m-shape ordered set partitions, square array read by ascending antidiagonals, A(m, n) for m, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 9, 1, 1, 7, 13, 35, 1, 1, 21, 121, 75, 161, 1, 1, 71, 1849, 3907, 541, 913, 1, 1, 253, 35641, 426405, 202741, 4683, 6103, 1, 1, 925, 762763, 65782211, 203374081, 15430207, 47293, 47319, 1, 1, 3433, 17190265, 11872636325, 323213457781, 173959321557

Offset: 1

Peter Luschny, Aug 02 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.
If m = 0, all possible sizes are zero. Thus the number of ordered set partitions of 0-shape is the number of ordered partitions of n (partition numbers A101880).
If m = 1, the set is {1, 2, ..., n} and the set of all possible sizes are the integer partitions of n. Thus the number of ordered set partitions of 1-shape is a Fubini number (sequence A000670).
If m = 2, the set is {1, 2, ..., 2n} and the number of ordered set partitions of 2-shape is also the number of 2-packed words of degree n (sequence A094088).

			[ n ] [0  1   2      3         4            5                  6]
[ m ] -----------------------------------------------------------
[ 0 ] [1, 1,  3,     9,       35,          161,              913]  A101880
[ 1 ] [1, 1,  3,    13,       75,          541,             4683]  A000670
[ 2 ] [1, 1,  7,   121,     3907,       202741,         15430207]  A094088
[ 3 ] [1, 1, 21,  1849,   426405,    203374081,     173959321557]  A243664
[ 4 ] [1, 1, 71, 35641, 65782211, 323213457781, 3482943541940351]  A243665
        A244174
For example the number of ordered set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] is 1, 168 and 1680 respectively. Thus A(3,3) = 1849.
Formatted as a triangle:
[1]
[1, 1]
[1, 1, 3]
[1, 1, 3, 9]
[1, 1, 7, 13, 35]
[1, 1, 21, 121, 75, 161]
[1, 1, 71, 1849, 3907, 541, 913]
[1, 1, 253, 35641, 426405, 202741, 4683, 6103]

Without order: A260876.

Cf. A000670, A094088, A101880, A243664, A243665, A243666, A244174.

def A260883(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
for m in (0..4): print([A260883(m, n) for n in (0..6)])

From Petros Hadjicostas, Aug 02 2019: (Start)

Conjecture: For n >= 0, let P be the set of all possible lists (a_1, ..., a_n) of nonnegative integers such that a_1*1 + a_2*2 + ... + a_n*n = n. Consider terms of the form multinomial(n*m, m*[1,..., 1, 2,..., 2,..., n,..., n]) * multinomial(a_1 + ... + a_n, [a_1,..., a_n]), where in the list [1,..., 1, 2,..., 2,..., n,..., n] the number 1 occurs a_1 times, 2 occurs a_2 times, ..., and n occurs a_n times. (Here a_n = 0 or 1.) Summing these terms over P we get A(m, n) provided m >= 1. (End)

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

Original entry on oeis.org

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)])

A309522 Generalized Blasius numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 4, 6, 14, 1, 1, 11, 34, 24, 42, 1, 1, 36, 375, 496, 120, 132, 1, 1, 127, 6306, 27897, 11056, 720, 429, 1, 1, 463, 129256, 3156336, 3817137, 349504, 5040, 1430, 1, 1, 1717, 2877883, 514334274, 3501788976, 865874115, 14873104, 40320, 4862

Offset: 0

Petros Hadjicostas and Peter Luschny, Aug 06 2019

The generalized Blasius o.d.e. of order n whose infinite series solution involves row n of this square array appears in Salié (1955). Rows n = 2 and n = 3 of this array appear in Kuba and Panholzer (2014, 2016), who give combinatorial interpretations to the numbers in those two rows.

Eq. (22) in Kuba and Panholzer (2014, p. 23) and Eq. (5) in Kuba and Panholzer (2016, p. 233) are general o.d.e.s for generating infinite sequences of numbers with some combinatorial properties. Even though they bear some similarity to Salié's general o.d.e., it is not clear whether either one can be used to give combinatorial interpretation to the numbers in rows n >= 4 of the current square array.

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
[0] 1, 1,  2,    5,      14,         42,           132, ...  A000108
[1] 1, 1,  2,    6,      24,        120,           720, ...  A000142
[2] 1, 1,  4,   34,     496,      11056,        349504, ...  A002105
[3] 1, 1, 11,  375,   27897,    3817137,     865874115, ...  A018893
[4] 1, 1, 36, 6306, 3156336, 3501788976, 7425169747776, ...
     A260878|

Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37; see p. 8. [This article was based on his PhD thesis. He corrected c_6 = A(n=3, k=6) but his "correction" of c_7 = A(n=3, k=7) was not right!]
Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37 [English translation by J. Vanier on behalf of the National Advisory Committee for Aeronautics (NACA), 1950]; see p. 8. [This is a translation of Blasius' article. The value of c_6 = A(n=3, k=6) was corrected in the article and the translation, but the "correction" for c_7 = A(n=3, k=7) in both documents is wrong.]
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, Discrete Mathematics 339(1) (2016), 227-254.
Hans Salié, Über die Koeffizienten der Blasiusschen Reihen, Math. Nachr. 14 (1955), 241-248 (1956). [In the article the array is denoted by c^{(n)}_v for n, v >= 1. We have A(n, k) = c^{(n)}_{k+1} for n >= 1 and k >= 0. The Catalan numbers (row n = 0 for A(n, k)) do not appear in Salié's article.]

Rows include A000108, A000142, A002105 (shifted), A018893.
Columns include A260878.
Cf. A256522 (Blasius constant), A260876.

A := proc(n, k) option remember; if k = 0 then 1 else
add(binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v), v=0..k-1) fi end:
seq(seq(A(n-k, k), k=0..n), n=0..9);

A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k - 1, n*v]*A[n, v]* A[n, k - 1 - v], {v, 0, k - 1}]];
Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)

A(n, k) = Sum_{v=0..k-1} binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v) for k > 0 and A(n, 0) = 1.
A(n, 2) = A260876(n, 2) = binomial(2*n - 1, n) + 1 for n >= 0.
A(n, 3) = A260876(n, 2) + A260876(n, 3) - 1 = (binomial(3*n - 1, 2*n) + 1) * (binomial(2*n - 1, n) + 1) + binomial(3*n - 1, n) for n >= 1.

A309725 Number of set partitions of {1,2,...,3n} with sizes in {[n, n, n], [2n, n], [3n]}.

Original entry on oeis.org

3, 5, 31, 365, 6271, 129130, 2877421, 66628441, 1578320767, 37983592076, 925196176906, 22754692780561, 564123212097901, 14079691134569845, 353428830512017081, 8915830309096530865, 225890912989184760703, 5744976464242932324976, 146603288011226858621356

Offset: 0

Peter Luschny, Aug 14 2019

Fourth column in A260876. Apart from a(0) the same as A326998.

Cf. A260876, A260878, A326998.

A327003 Irregular triangle read by rows in which the n-th row lists multinomials for partitions of 3n which have only parts which are multiples of 3, in Hindenburg order.

Original entry on oeis.org

1, 1, 1, 10, 1, 84, 280, 1, 220, 462, 9240, 15400, 1, 455, 5005, 50050, 210210, 1401400, 1401400, 1, 816, 18564, 185640, 24310, 4084080, 13613600, 2858856, 85765680, 285885600, 190590400, 1, 1330, 54264, 542640, 293930, 24690120, 82300400, 32332300, 135795660, 2715913200, 4526522000, 3802278480, 38022784800, 76045569600, 36212176000

Offset: 0

Peter Luschny, Aug 14 2019

The Hindenburg order refers to the partition generating algorithm of C. F. Hindenburg (1779). [Knuth 7.2.1.4H]

			The irregular triangle starts:
[0] [1]
[1] [1]
[2] [1, 10]
[3] [1, 84, 280]
[4] [1, 220, 462, 9240, 15400]
[5] [1, 455, 5005, 50050, 210210, 1401400, 1401400]
[6] [1, 816, 18564, 185640, 24310, 4084080, 13613600, 2858856, 85765680, 285885600, 190590400]

Cf. A000012 (m=0, subdivided into rows of length A000041), A080575 (m=1), A257490 (m=2), this sequence (m=3), A327004 (m=4).
Cf. A000041 (length of rows), A291973 (sum of rows), A291451 (coarser subdivision).
Cf. A260876.

def A327003row(n):
    shapes = ([3*x for x in p] for p in Partitions(n))
    return [SetPartitions(sum(s), s).cardinality() for s in shapes]
for n in (0..7): print(A327003row(n))

Row of lengths are in A000041.

A327004 Irregular triangle read by rows in which the n-th row lists multinomials for partitions of 4n which have only parts which are multiples of 4, in Hindenburg order.

Original entry on oeis.org

1, 1, 1, 35, 1, 495, 5775, 1, 1820, 6435, 450450, 2627625, 1, 4845, 125970, 4408950, 31177575, 727476750, 2546168625, 1, 10626, 735471, 25741485, 1352078, 1338557220, 15616500900, 1577585295, 165646455975, 1932541986375, 4509264634875

Offset: 0

Peter Luschny, Aug 14 2019

The Hindenburg order refers to the partition generating algorithm of C. F. Hindenburg (1779). [Knuth 7.2.1.4H]

			The irregular triangle starts:
[0] [1]
[1] [1]
[2] [1, 35]
[3] [1, 495, 5775]
[4] [1, 1820, 6435, 450450, 2627625]
[5] [1, 4845, 125970, 4408950, 31177575, 727476750, 2546168625]
[6] [1, 10626, 735471, 25741485, 1352078, 1338557220, 15616500900, 1577585295, 165646455975, 1932541986375, 4509264634875]

Cf. A000012 (m=0, subdivided into rows of length A000041), A080575 (m=1), A257490 (m=2), A327003 (m=3), this sequence (m=4).
Cf. A000041 (length of rows), A291975 (sum of rows), A291452 (coarser subdivision).
Cf. A260876.

def A327004row(n):
    shapes = ([4*x for x in p] for p in Partitions(n))
    return [SetPartitions(sum(s), s).cardinality() for s in shapes]
for n in (0..6): print((A327004row(n)))

Row of lengths are in A000041.

cp's OEIS Frontend

A326996 Main diagonal of A260876.

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A260883 Number of m-shape ordered set partitions, square array read by ascending antidiagonals, A(m, n) for m, n >= 0.

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

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A309522 Generalized Blasius numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.

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A309725 Number of set partitions of {1,2,...,3n} with sizes in {[n, n, n], [2n, n], [3n]}.

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