A272644 -id:A272644 - cp's OEIS frontend

A297195 Number of bitriangular permutations (row sums of A272644 if that triangle is prefixed with two rows for n=0,1).

1, 0, 1, 2, 7, 28, 133, 726, 4483, 30896, 235105, 1957930, 17712799, 172980804, 1813760317, 20323234814, 242353047355, 3064550705752, 40958281206169, 576917769130578, 8541793624670551, 132623408805525740, 2154730841214003061, 36560670776303600422, 646697046042017004787

Offset: 0

N. J. A. Sloane, Jan 10 2018

Define c(n) = Sum_{m=2..n-1} C(n-1, m-1)^(-2). Define b(1) = x and b(n+1) = b(n) + (Sum_{m=2..n-1} b(m)*b(n+1-m)*C(n-1, m-1)^(-2))/n^2 for n>0. Then b(n) is a polynomial in x and so is (b(n+1)-b(n))/x^2 whose constant term is c(n)/n^2. The Hone et.al.[2002] link denotes x with alpha_2 and alpha_k = (k-1)!^2*b(k). Conjecture: Asymptotic expansion of c(n) = 2*Sum_{i>1} a(i)/n^i. - Michael Somos, Oct 17 2024

			G.f. = 1 + x^2 + 2*x^3 + 7*x^4 + 28*x^5 + 133*x^6 + 726*x^7 + ... - _Michael Somos_, Oct 17 2024

A.N.W. Hone, N. Joshi and A.V. Kitaev, An Entire Function Defined by a Nonlinear Recurrence Relation, J. of the London Math. Soc., Oct. 2002, vol. 66, iss. 2, pp. 377-387.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]

Cf. A272644.

A297195 := proc(n)
    add(A272644(n, m), m=0..n) ;
end proc:
seq(A297195(n), n=0..30) ; # R. J. Mathar, Mar 04 2018

A272644[n_, m_] := Sum[StirlingS2[m+1, i+1] (-1)^(m-i) i^(n-m) i!, {i, 0, m}];
a[n_] := If[n == 1, 1, Sum[A272644[n, m], {m, 1, n-1}]];
Array[a, 24] (* Jean-François Alcover, Apr 03 2020 *)

{a(n) = if(n<2, n==0, sum(m=1, n-1, sum(i=0, m, (-1)^(m-i)*i^(n-m)*i!*stirling(m+1, i+1, 2))))}; /* Michael Somos, Oct 17 2024 */

Some terms corrected by Alois P. Heinz, Oct 17 2024

A272645 a(n) = largest term in row n of array in A272644.

Original entry on oeis.org

1, 1, 5, 13, 73, 301, 2069, 11581, 95401, 673261, 6487445, 55213453, 610093513, 6077248381, 75796724309, 864806272861, 12020754177001, 154546274524621, 2369364111428885, 33888536448984493, 568128719132038153, 8947078682269788061

Offset: 2

N. J. A. Sloane, May 07 2016

Gheorghe Coserea, Table of n, a(n) for n = 2..101
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.

Cf. A272644.

A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
apply(vecmax, vector(22, n, vector(n, m, A(n+1, m))))  \\ Gheorghe Coserea, May 16 2016

See A272644.

More terms from Gheorghe Coserea, May 16 2016

A048144 a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.

Original entry on oeis.org

1, 1, 5, 73, 2069, 95401, 6487445, 610093513, 75796724309, 12020754177001, 2369364111428885, 568128719132038153, 162835627057766030549, 54975855375379966645801, 21593185551426744571090325, 9762238510837560633366673993, 5033241437347149354018370856789

Offset: 0

N. J. A. Sloane

Number of digraphs with loops, with labeled vertices and labeled arcs, with n arcs and with no vertex of indegree 0 or outdegree 0, cf. A121936, A122418, A122399. - Vladeta Jovovic, Sep 06 2006
Chromatic invariant of the complete bipartite graph K_{n+1,n+1}. - Eric W. Weisstein, Jul 11 2011
Generally, for p >= 1, Sum_{k=0..n} (k!*StirlingS2(n,k))^p is asymptotic to n^(p*n+1/2) * sqrt(Pi/(2*p*(1-log(2))^(p-1))) / (exp(p*n) * log(2)^(p*n+1)). - Vaclav Kotesovec, May 10 2014

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Cf. A000670, A242280, A212084, A120732, A104602, A272644, A371761.

a := proc(n) local A, j; A := proc(n, k) option remember; if n = 0 then n^k else add(binomial(k + `if`(j>0, 1, 0), j+1) * A(n-1, k-j), j = 0..k) fi end: A(n,n) end:
seq(a(n), n = 0..16);  # Peter Luschny, Nov 20 2024

Table[Sum[(k!)^2*StirlingS2[n,k]^2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)

a(n) = sum(k=0, n, k!^2*stirling(n, k, 2)^2); \\ Michel Marcus, Mar 07 2020

from functools import cache
from math import comb as binomial
@cache
def A(n, k): return int(k == 0) if n == 0 else sum(binomial(k + int(j > 0), j + 1) * A(n - 1, k - j) for j in range(k + 1))
a = lambda n: A(n, n)
print([a(n) for n in range(17)])  # Peter Luschny, Nov 20 2024

E.g.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(exp(j*x)-1)^n. a(n) = Sum_{k=0..n} Stirling2(n,k)*k!*A104602(k). - Vladeta Jovovic, Mar 25 2006
a(n) ~ sqrt(Pi/(1-log(2))) * n^(2*n+1/2) / (2*exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, May 09 2014
E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n * exp(-n*x). - Paul D. Hanna, Mar 26 2018
E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n * exp(-n*(n+1)*x). - Paul D. Hanna, Mar 26 2018
a(n) = A272644(2n,n). - Alois P. Heinz, Oct 17 2024
a(n) = A371761(n, n). - Peter Luschny, Nov 20 2024
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x + y)). - Ilya Gutkovskiy, Apr 24 2025

A371761 Array read by antidiagonals: The number of parades with n girls and k boys that begin with a girl and end with a boy.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 5, 1, 0, 0, 1, 13, 13, 1, 0, 0, 1, 29, 73, 29, 1, 0, 0, 1, 61, 301, 301, 61, 1, 0, 0, 1, 125, 1081, 2069, 1081, 125, 1, 0, 0, 1, 253, 3613, 11581, 11581, 3613, 253, 1, 0, 0, 1, 509, 11593, 57749, 95401, 57749, 11593, 509, 1, 0

Offset: 0

Peter Luschny, Apr 05 2024

The name derives from a proposition by Donald Knuth, who describes the setup of a "girls and boys parade" as follows: "There are m girls {g_1, ..., g_m} and n boys {b_1, ..., b_n}, where g_i is younger than g_{i+1} and b_j is younger than b_{j+1}, but we know nothing about the relative ages of g_i and b_j. In how many ways can they all line up in a sequence such that no girl is directly preceded by an older girl and no boy is directly preceded by an older boy?" [Our notation: A <- D, n <- m, k <- n].
In A344920, the Worpitzky transform is defined as a sequence-to-sequence transformation WT := A -> B, where B(n) = Sum_{k=0..n} A163626(n, k)*A(k). (If A(n) = 1/(n + 1) then B(n) are the Bernoulli numbers (with B(1) = 1/2.)) The rows of the array are the Worpitzky transforms of the powers up to the sign (-1)^k.
The array rows are recursively generated by applying the Akiyama-Tanigawa algorithm to the powers (see the Python implementation below). In this way the array becomes the image of A004248 under the AT-transformation when applied to the rows of A004248. This makes the array closely linked to A344499, which is generated in the same way, but applied to the columns of A004248.
Conjecture: Row n + 1 is row 2^n in table A136301, where a probabilistic interpretation is given (see the link to Parsonnet's paper below).

			Array starts:
[0] 1, 0,   0,     0,      0,       0,        0,         0,          0, ...
[1] 0, 1,   1,     1,      1,       1,        1,         1,          1, ...
[2] 0, 1,   5,    13,     29,      61,      125,       253,        509, ...
[3] 0, 1,  13,    73,    301,    1081,     3613,     11593,      36301, ...
[4] 0, 1,  29,   301,   2069,   11581,    57749,    268381,    1191989, ...
[5] 0, 1,  61,  1081,  11581,   95401,   673261,   4306681,   25794781, ...
[6] 0, 1, 125,  3613,  57749,  673261,  6487445,  55213453,  431525429, ...
[7] 0, 1, 253, 11593, 268381, 4306681, 55213453, 610093513, 6077248381, ...
.
Seen as triangle T(n, k) = A(n - k, k):
  [0] 1;
  [1] 0, 0;
  [2] 0, 1,  0;
  [3] 0, 1,  1,   0;
  [4] 0, 1,  5,   1,   0;
  [5] 0, 1, 13,  13,   1,  0;
  [6] 0, 1, 29,  73,  29,  1, 0;
  [7] 0, 1, 61, 301, 301, 61, 1, 0;
.
A(n, k) as sum of powers:
  A(2, k) =  -3+   2*2^k;
  A(3, k) =   7-  12*2^k+    6*3^k;
  A(4, k) = -15+  50*2^k-   60*3^k+   24*4^k;
  A(5, k) =  31- 180*2^k+  390*3^k-  360*4^k+  120*5^k;
  A(6, k) = -63+ 602*2^k- 2100*3^k+ 3360*4^k- 2520*5^k+  720*6^k;
  A(7, k) = 127-1932*2^k+10206*3^k-25200*4^k+31920*5^k-20160*6^k+5040*7^k;

Peter Luschny, Table of n, a(n) for antidiagonals 0..140.
Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
Beáta Bényi and Péter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n, k}.
Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (16.2).
Brian Parsonnet, Probability of Derangements.

Variant: A272644.
Rows include: A344920 (row 2, signed), A006230 (row 3).
Row sums of triangle (n>=2): A297195, alternating row sums: A226158.
Diagonal of array: A048144.
Cf. A004248, A075263, A028246, A173018, A136301, A371762, A136126 (similar), A099594, A344499.

egf := 1/(exp(w) + exp(z) - exp(w + z)): serw := n -> series(egf, w, n + 1):
# Returns row n (>= 0) with length len (> 0):
R := n -> len -> local k;
seq(k!*coeff(series(n!*coeff(serw(n), w, n), z, len), z, k), k = 0..len - 1):
seq(lprint(R(n)(9)), n = 0..7);
# Explicit with Stirling2 :
A := (n, k) -> local j; add(j!^2*Stirling2(n, j)*Stirling2(k, j), j = 0..min(n, k)): seq(lprint(seq(A(n, k), k = 0..8)), n = 0..7);
# Using the unsigned Worpitzky transform.
WT := (a, len) -> local n, k;
seq(add((-1)^(n - k)*k!*Stirling2(n + 1, k + 1)*a(k), k=0..n), n = 0..len-1):
Arow := n -> WT(x -> x^n, 8): seq(lprint(Arow(n)), n = 0..8);
# Two recurrences:
A := proc(n, k) option remember; local j; if k = 0 then return k^n fi;
add(binomial(n, j)*(A(n-j, k-1) + A(n-j+1, k-1)), j = 1..n) end:
A := proc(n, k) option remember; local j; if n = 0 then 0^k else
add(binomial(k + `if`(j=0,0,1), j+1)*A(n-1, k-j), j = 0..k) fi end:

(* Using the unsigned Worpitzky transform. *)
Unprotect[Power]; Power[0, 0] = 1;
W[n_, k_] := (-1)^(n - k) k! StirlingS2[n + 1, k + 1];
WT[a_, len_] := Table[Sum[W[n, k] a[k], {k, 0, n}], {n, 0, len-1}];
A371761row[n_, len_] := WT[#^n &, len];
Table[A371761row[n, 9], {n, 0, 8}] // MatrixForm
(* Row n >= 1 by linear recurrence: *)
RowByLRec[n_, len_] := LinearRecurrence[Table[-StirlingS1[n+1, n+1-k], {k, 1, n}],
A371761row[n, n+1], len]; Table[RowByLRec[n, 9], {n, 1, 8}] // MatrixForm

from functools import cache
from math import comb as binomial
@cache
def A(n, k):
    if n == 0: return int(k == 0)
    return sum(binomial(k + int(j > 0), j + 1) * A(n - 1, k - j)
           for j in range(k + 1))
for n in range(8): print([A(n, k) for k in range(8)])

# The Akiyama-Tanigawa algorithm for powers generates the rows.
def ATPowList(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = k**n   # Changing this to R[k] = (n + 1)**k generates A344499.
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(8): print([n], ATPowList(n, 9))

def A371761(n, k): return sum((-1)^(j - k) * factorial(j) * stirling_number2(k + 1, j + 1) * j^n for j in range(k + 1))
for n in range(9): print([A371761(n, k) for k in range(8)])

A(n, k) = k! * [z^k] (n! * [w^n] 1/(exp(w) + exp(z) - exp(w + z))).
A(n, k) = k! * [w^k] (Sum_{j=0..n} A075263(n, n - j) * exp(j*w)).
A(n, k) = Sum_{j=0..k} (-1)^(j-k) * Stirling2(k + 1, j + 1) * j! * j^n.
A(n, k) = Sum_{j=0..min(n,k)} (j!)^2 * Stirling2(n, j) * Stirling2(k, j).
A(n, k) = Sum_{j=0..n} (-1)^(n-j)*A028246(n, j) * j^k; this is explicit:
A(n, k) = Sum_{j=0..n} Sum_{m=0..n} binomial(n-m, n-j) * Eulerian1(n, m) * j^k *(-1)^(n-j), where Eulerian1 = A173018.
A(n, k) = Sum_{j=0..k} binomial(k + [j>0], j+1)*A(n-1, k-j) for n > 0.
A(n, k) = Sum_{j=1..n} binomial(n, j)*(A(n-j, k-1) + A(n-j+1, k-1)) for n,k >= 1.
Row n (>=1) satisfies a linear recurrence:
A(n, k) = -Sum_{j=1..n} Stirling1(n + 1, n + 1 - j)*A(n, k - j) if k > n.
A(n, k) = [x^k] (Sum_{j=0..n} A371762(n, j)*x^j) / (Sum_{j=0..n} Stirling1(n + 1, n + 1 - j)*x^j).
A(n, k) = A(k, n). (From the symmetry of the bivariate exponential g.f.)
Let T(n, k) = A(n - k, k) and G(n) = Sum_{k=0..n} (-1)^k*T(n, k) the alternating row sums of the triangle. Then G(n) = (n + 2)*Euler(n + 1, 1) and as shifted Genocchi numbers G(n) = -2*(n + 2)*PolyLog(-n - 1, -1) = -A226158(n + 2).

A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1

Offset: 2

Emeric Deutsch, Jan 14 2005

Row n has n - 1 terms. Row sums yield the Fine numbers (A000957).
Related to the number of certain sets of non-crossing partitions for the root system A_n (p. 11, Athanasiadis and Savvidou). - Tom Copeland, Oct 19 2014
T(n,k) is the number of permutations pi of [n-1] with k - 1 descents such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018
The absolute values of the polynomials at -1 and j (cube root of 1) seem to be given by A126120 and A005043. - F. Chapoton, Nov 16 2021
Don Knuth observes that this sequence also arrises from the enumeration of restricted max-and-min-closed relations, only there it appears as an array read by antidiagonals: see the Knuth "Notes" link and A372068. Knuth also gives a formula expressing the array A372368 in terms of this array. He also reports that there is strong experimental evidence that the n-th term of row m in this array is a polynomial of degree 2*m-2 in n. - N. J. A. Sloane, May 12 2024

			T(4, 2) = 4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U = (1, 1), D = (1,-1) and * indicates the peaks.
Triangle starts:
   1;
   1,  1;
   1,  4,   1;
   1,  8,   8,    1;
   1, 13,  29,   13,    1;
   1, 19,  73,   73,   19,    1;
   1, 26, 151,  266,  151,   26,    1;
   1, 34, 276,  749,  749,  276,   34,   1;
   1, 43, 463, 1781, 2762, 1781,  463,  43,  1;
   1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1;
   ...
As an array (for which the rows of the preceding triangle are the antidiagonals):
   1,  1,    1,     1,      1,      1,       1,        1,        1, ...
   1,  4,    8,    13,     19,     26,      34,       43,       53, ...
   1,  8,   29,    73,    151,    276,     463,      729,     1093, ...
   1, 13,   73,   266,    749,   1781,    3758,     7253,    13061, ...
   1, 19,  151,   749,   2762,   8321,   21659,    50471,   107833, ...
   1, 26,  276,  1781,   8321,  31004,   97754,   271125,   679355, ...
   1, 34,  463,  3758,  21659,  97754,  367285,  1196665,  3478915, ...
   1, 43,  729,  7253,  50471, 271125, 1196665,  4526470, 15118415, ...
   1, 53, 1093, 13061, 107833, 679355, 3478915, 15118415, 57500480, ...
   ...

Alois P. Heinz, Rows n = 2..142, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A000957, A108263, A132081.

See also A099594, A272644, A372066, A372067, A372068.

T := (n, k) -> add((j/(n-j))*binomial(n-j, k-j)*binomial(n-j,k), j=0..min(k,n-k)): for n from 2 to 13 do seq(T(n, k), k = 1..n-1) od; # yields the sequence in triangular form

T[n_, k_] := Sum[(j/(n-j))*Binomial[n-j, k-j]*Binomial[n-j, k], {j, 0, Min[k, n-k]}]; Table[T[n, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n, k) = Sum_{j=0..min(k, n-k)} (j/(n-j)) * C(n-j, k-j) * C(n-j, k), n >= 2.
G.f.: t*z*r/(1 - t*z*r), where r = r(t, z) is the Narayana function defined by r = z*(1 + r)*(1 + t*r).
From Tom Copeland, Oct 19 2014: (Start)
With offset 0 for A108263 and offset 1 for A132081, row polynomials of this entry P(n, x) = Sum_{i} A108263(n, i)*x^i*(1 + x)^(n - 2*i) = Sum_{i} A132081(n - 2, i)*x^i*(1 + x)^(n - 2*i).
E.g., P(4, x) = 1*x*(1 + x)^(4 - 2*1) + 2*x^2*(1 + x)^(4 - 2*2) = x + 4*x^2 + x^3.
Equivalently, let Q(n, x) be the row polynomials of A108263. Then P(n, x) = (1 + x)^n * Q(n, x/(1 + x)^2).
E.g., P(4, x) = (1 + x)^4 * (x/(1 + x)^2 + 2 * (x/(1 + x)^2)^2).
See Athanasiadis and Savvidou (p. 7). (End)

A382740 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 43, 127, 43, 1, 1, 91, 559, 559, 91, 1, 1, 187, 2071, 4327, 2071, 187, 1, 1, 379, 7039, 25831, 25831, 7039, 379, 1, 1, 763, 22807, 133783, 233551, 133783, 22807, 763, 1, 1, 1531, 71839, 636679, 1748791, 1748791, 636679, 71839, 1531, 1

Offset: 1

Seiichi Manyama, Apr 04 2025

			Square array begins:
  1,   1,    1,      1,       1,        1, ...
  1,   7,   19,     43,      91,      187, ...
  1,  19,  127,    559,    2071,     7039, ...
  1,  43,  559,   4327,   25831,   133783, ...
  1,  91, 2071,  25831,  233551,  1748791, ...
  1, 187, 7039, 133783, 1748791, 18207367, ...

Cf. A272644, A382741, A382742.

Cf. A382734.

a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 2))/2;

E.g.f.: (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).
A(n,k) = A(k,n).
A(n,k) = (1/2) * A382734(n,k).

A382741 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/3) * (1 / (exp(x) + exp(y) - exp(x+y))^3 - 1).

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 25, 25, 1, 1, 57, 193, 57, 1, 1, 121, 889, 889, 121, 1, 1, 249, 3361, 7593, 3361, 249, 1, 1, 505, 11545, 47641, 47641, 11545, 505, 1, 1, 1017, 37633, 253737, 465601, 253737, 37633, 1017, 1, 1, 2041, 118969, 1228249, 3657721, 3657721, 1228249, 118969, 2041, 1

Offset: 1

Seiichi Manyama, Apr 04 2025

			Square array begins:
  1,   1,     1,      1,       1,        1, ...
  1,   9,    25,     57,     121,      249, ...
  1,  25,   193,    889,    3361,    11545, ...
  1,  57,   889,   7593,   47641,   253737, ...
  1, 121,  3361,  47641,  465601,  3657721, ...
  1, 249, 11545, 253737, 3657721, 40666089, ...

Cf. A272644, A382740, A382742.

Cf. A382735.

a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n, j, 2)*stirling(k, j, 2))/3;

E.g.f.: (1/3) * (1 / (exp(x) + exp(y) - exp(x+y))^3 - 1).
A(n,k) = A(k,n).
A(n,k) = (1/3) * A382735(n,k).

A372068 Array read by antidiagonals: T(m,n) (m >= 0, n >= 0) = number of max-and-min-closed constraints between an m-element set and an n-element set.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 13, 8, 1, 1, 16, 38, 38, 16, 1, 1, 32, 104, 147, 104, 32, 1, 1, 64, 272, 506, 506, 272, 64, 1, 1, 128, 688, 1612, 2103, 1612, 688, 128, 1, 1, 256, 1696, 4856, 7887, 7887, 4856, 1696, 256, 1, 1, 512, 4096, 14016, 27477, 34088, 27477, 14016, 4096, 512, 1

Offset: 0

N. J. A. Sloane, May 12 2024, based on emails from Don Knuth, May 06 2024 and May 08 2024

See the Knuth "Notes" link for much more information about these sequences. The present sequence is called "tab" in Part 2 of the Notes.

			The initial antidiagonals are:
   1,
   1, 1,
   1, 2, 1,
   1, 4, 4, 1,
   1, 8, 13, 8, 1,
   1, 16, 38, 38, 16, 1,
   1, 32, 104, 147, 104, 32, 1,
   1, 64, 272, 506, 506, 272, 64, 1,
   1, 128, 688, 1612, 2103, 1612, 688, 128, 1,
   1, 256, 1696, 4856, 7887, 7887, 4856, 1696, 256, 1,
   1, 512, 4096, 14016, 27477, 34088, 27477, 14016, 4096, 512, 1,
   ...
The array begins:
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
   1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, ...
   1, 8, 38, 147, 506, 1612, 4856, 14016, 39104, 106112, ...
   1, 16, 104, 506, 2103, 7887, 27477, 90498, 285072, 865856, ...
   1, 32, 272, 1612, 7887, 34088, 134825, 498465, 1746830, 5859404, ...
   1, 64, 688, 4856, 27477, 134825, 597539, 2451038, 9455182, 34687916, ...
   1, 128, 1696, 14016, 90498, 498465, 2451038, 11055950, 46570858, 185484836, ...
   1, 256, 4096, 39104, 285072, 1746830, 9455182, 46570858, 212833803, 914854829, ...
   1, 512, 9728, 106112, 865856, 5859404, 34687916, 185484836, 914854829, 4223468802, ...
   ...

D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A099594, A100754, A272644, A372066, A372067.

cp's OEIS Frontend

A297195 Number of bitriangular permutations (row sums of A272644 if that triangle is prefixed with two rows for n=0,1).

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A272645 a(n) = largest term in row n of array in A272644.

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A048144 a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.

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A371761 Array read by antidiagonals: The number of parades with n girls and k boys that begin with a girl and end with a boy.

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A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

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A382740 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).

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A382741 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/3) * (1 / (exp(x) + exp(y) - exp(x+y))^3 - 1).

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