A000680 -id:A000680 - cp's OEIS frontend

A154283 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(2n+1,i) binomial(k+2-i,2)^n, 0 <= k <= 2*(n-1).

1, 1, 4, 1, 1, 20, 48, 20, 1, 1, 72, 603, 1168, 603, 72, 1, 1, 232, 5158, 27664, 47290, 27664, 5158, 232, 1, 1, 716, 37257, 450048, 1822014, 2864328, 1822014, 450048, 37257, 716, 1, 1, 2172, 247236, 6030140, 49258935, 163809288, 242384856, 163809288, 49258935, 6030140, 247236, 2172, 1

Offset: 1

Roger L. Bagula, Jan 06 2009

From Yahia Kahloune, Jan 30 2014: (Start)
In general, let b(k,e,p) = Sum_{i=0..k} (-1)^i*binomial(e*p+1,i)*binomial(k+e-i,e)^p. Then T(n,k) = b(k,2,n).
With these coefficients we can calculate: Sum_{i=1..n} binomial(i+e-1,e)^p = Sum_{k=0..e*(p-1)} b(k,e,p)*binomial(n+e+k,e*p+k).
For example, A085438(n) = Sum_{i=1..n} binomial(1+i,2)^3 = T(3,0)*binomial(2+n,7) + T(3,1)*binomial(3+n,7) + T(3,2)*binomial(4+n,7) + T(3,3)*binomial(5+n,7) + T(3,4)*binomial(6+n,7) = (1/5040)*(90*n^7 + 630*n^6 + 1638*n^5 + 1890*n^4 + 840*n^3 - 48*n).
(End)
T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n with exactly k descents. A descent is a pair of adjacent elements with the second element less than the first. - Andrew Howroyd, May 06 2020

			Triangle begins:
  1;
  1,     4,       1;
  1,    20,      48,        20,           1;
  1,    72,     603,      1168,         603,           72,           1;
  1,   232,    5158,     27664,       47290,        27664,        5158,  232, 1;
  1,   716,   37257,    450048,     1822014,      2864328,     1822014, ...;
  1,  2172,  247236,   6030140,    49258935,    163809288,   242384856, ...;
  1,  6544, 1568215,  72338144,  1086859301,   6727188848, 19323413187, ...;
  1, 19664, 9703890, 811888600, 21147576440, 225167210712, ... ;
  ...
The T(2,1) = 4 permutations of 1122 with 1 descent are 1212, 1221, 2112, 2211. - _Andrew Howroyd_, May 15 2020

Andrew Howroyd, Table of n, a(n) for n = 1..1600 (rows 1..40)
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 12.
Helmut Prodinger, On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs , arXiv:1102.5186 [math.CO], 2011.

Columns k=0..9 are A000012, A061981, A151624, A151625, A151626, A151627, A151628, A151629, A151630, A151631.
Row sums are A000680.
Similar triangles for e=1..6: A173018 (or A008292), this sequence, A174266, A236463, A237202, A237252.
Cf. A000680, A005799, A085438, A334781.

[(&+[(-1)^j*Binomial(2*n+1,j)*Binomial(k-j+2,2)^n: j in [0..k]]): k in [0..2*n-2], n in [1..12]]; // G. C. Greubel, Jun 13 2022

A154283 := proc(n,k)
        (1-x)^(2*n+1)*add( (l*(l+1)/2)^n*x^(l-1),l=0..k+1) ;
        coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Feb 01 2013

p[x_, n_]= (1-x)^(2*n+1)*Sum[(k*(k+1)/2)^n*x^k, {k, 0, Infinity}]/x;
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n,10}]//Flatten

T(n,k)={sum(i=0, k, (-1)^i*binomial(2*n+1, i)*binomial(k+2-i, 2)^n)} \\ Andrew Howroyd, May 09 2020

def A154283(n,k): return sum((-1)^j*binomial(2*n+1, j)*binomial(k-j+2, 2)^n for j in (0..k))
flatten([[A154283(n,k) for k in (0..2*n-2)] for n in (1..12)]) # G. C. Greubel, Jun 13 2022

T(n,k) = (-1) times coefficient of x^k in (x-1)^(2*n+1) * Sum_{k>=0} (k*(k+1)/2)^n *x^(k-1).
From Yahia Kahloune, Jan 29 2014: (Start)
Sum_{i=1..n} binomial(1+i,2)^p = Sum_{k=0..2*p-2} T(p,k)*binomial(n+2+k,2*p+1).
binomial(n,2)^p = Sum_{k=0..2*p-2} T(p,k)*binomial(n+k,2*p). (End)
From Peter Bala, Dec 21 2019: (Start)
E.g.f. as a continued fraction: (1-x)/(1-x + ( 1-exp((1-x)^2*t))*x/(1-x + (1-exp(2*(1-x)^2*t))*x/(1-x + (1-exp(3*(1-x)^2*t))*x/(1-x + ... )))) =  1 + x*t + x*(x^2 + 4*x + 1)*t^2/2! + x*(x^4 + 20*x^3 + 48*x^2 + 20*x + 1)*t^3/3! + ... (use Prodinger equation 1.1).
The sequence of alternating row sums (unsigned) [1, 1, 2, 10, 104, 1816,...] appears to be A005799. (End)

Edited by N. J. A. Sloane, Jan 30 2014 following suggestions from Yahia Kahloune (among other things, the signs of all terms have been reversed).

Edited by Andrew Howroyd, May 09 2020

A132101 a(n) = (A001147(n) + A047974(n))/2.

Original entry on oeis.org

1, 1, 3, 11, 65, 513, 5363, 68219, 1016481, 17243105, 327431363, 6874989963, 158118876449, 3952936627361, 106729080101235, 3095142009014843, 95949394016339393, 3166329948046914369, 110821547820208233731, 4100397266856761733515

Offset: 0

Keith F. Lynch, Oct 31 2007

Also, number of distinct Tsuro tiles which are digonal in shape and have n points per side. Turning over is not allowed. See A132100 for definition and comments.
See the Burns et al. papers for another interpretation.
From Ross Drewe, Mar 16 2008: (Start)
This is also the number of arrangements of n pairs which are equivalent under the joint operation of sequence reversal and permutations of labels. Assume that the elements of n distinct pairs are labeled to show the pair of origin, e.g., [1 1], [2 2]. The number of distinguishable ways of arranging these elements falls as the conditions are made more general:
a(n) = A000680: element order is significant and the labels are distinguishable;
b(n) = A001147: element order is significant but labels are not distinguishable, i.e., all label permutations of a given sequence are equivalent;
c(n) = A132101: element order is weakened (reversal allowed) and all label permutations are equivalent;
d(n) = A047974: reversal allowed, all label permutations are equivalent and equivalence class maps to itself under joint operation.
Those classes that do not map to themselves form reciprocal pairs of classes under the joint operation and their number is r(n). Then c = b - r/2 = b - (b - d)/2 = (b+d)/2. A formula for r(n) is not available, but formulas are available for b(n) = A001147 and d(n) = A047974, allowing an explicit formula for this sequence.
c(n) is useful in extracting structure information without regard to pair ordering (see example). c(n) terms also appear in formulas related to binary operators, e.g., the number of binary operators in a k-valued logic that are invertible in 1 operation.
a(n) = (b(n) + c(n))/2, where b(n) = (2n)!/(2^n * n!) = A001147(n), c(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)! * k!) = A047974(n).
For 3 pairs, the arrangement A = [112323] is the same as B = [212133] under the permutation of the labels [123] -> [312] plus reversal of the elements, or vice versa. The unique structure common to A and B is {1 intact pair + 2 interleaved pairs}, where the order is not significant (contrast A001147). (End)

			a(2)=3 counts the arrangements [1122], [1212] and [1221]. - _R. J. Mathar_, Oct 18 2019

G. C. Greubel, Table of n, a(n) for n = 0..400
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts).
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche, and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section IV.

Cf. A000680, A001147, A047974, A007769, A054499.

Cf. A132100, A132101, A132102, A132103, A132104, A132105.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( (Exp(x+x^2) + 1/Sqrt(1-2*x))/2 ))); // G. C. Greubel, Jul 12 2024

A132101 := proc(n)
    (A001147(n)+A047974(n))/2 ;
end proc:
seq(A132101(n),n=0..30) ; # R. J. Mathar, Dec 20 2020

Table[((2n-1)!!+I^(-n)*HermiteH[n,I/2])/2,{n,0,30}] (* Jonathan Burns, Apr 05 2016 *)

[(factorial(n)*binomial(2*n,n) + (-2*i)^n*hermite(n,i/2))/2^(n+1) for n in range(31)] # G. C. Greubel, Jul 12 2024

D-finite with recurrence -(n-3)*a(n) +2*(n^2-3*n+1)*a(n-1) -(n-1)*a(n-2) -2*(2*n-5)*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Dec 20 2020

E.g.f.: (1/2)*( exp(x+x^2) + 1/sqrt(1-2*x) ). - G. C. Greubel, Jul 12 2024

Entry revised by N. J. A. Sloane, Nov 04 2011

A014609 a(n) = (5n)!/(5!^n).

Original entry on oeis.org

1, 1, 252, 756756, 11732745024, 623360743125120, 88832646059788350720, 28837919555681211870935040, 18975581770994682860770223800320, 23183587808948692737291767860055162880, 49120458506088132224064306071170476903628800

Offset: 0

BjornE (mdeans(AT)algonet.se)

nonn

a(n) is also the constant term in product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^5. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 18 2002

G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, 1998.

Alois P. Heinz, Table of n, a(n) for n = 0..105
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

Cf. A000680, A014606, A014608.

PARI
```
a(n)=if(n<0,0,(5*n)!/120^n)
```

A266735 Number of words on {1,1,2,2,3,3,...,n,n} avoiding the pattern 12345.

Original entry on oeis.org

1, 1, 6, 90, 2520, 102011, 5176504, 307027744, 20472135280, 1496594831506, 117857270562568, 9869468603141427, 870255083860881152, 80185525536941657225, 7673807618627318341436, 759049283017632212000140, 77292554293281131959377376, 8075621155990277422800518076

Offset: 0

N. J. A. Sloane, Jan 06 2016

nonn

Ferenc Balogh, A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length, preprint arXiv:1505.01389, 2015.
Shalosh B. Ekhad and Doron Zeilberger, The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r, 2014

Cf. A000680, A220097, A266734.

Column k=4 of A267479.

More terms from Alois P. Heinz, Jan 14 2016

A267479 Number A(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 6, 1, 0, 1, 1, 6, 43, 1, 0, 1, 1, 6, 90, 352, 1, 0, 1, 1, 6, 90, 1879, 3114, 1, 0, 1, 1, 6, 90, 2520, 47024, 29004, 1, 0, 1, 1, 6, 90, 2520, 102011, 1331664, 280221, 1, 0, 1, 1, 6, 90, 2520, 113400, 5176504, 41250519, 2782476, 1, 0

Offset: 0

Alois P. Heinz, Jan 15 2016

			Square array A(n,k) begins:
  1, 1,     1,       1,       1,       1,       1, ...
  0, 1,     1,       1,       1,       1,       1, ...
  0, 1,     6,       6,       6,       6,       6, ...
  0, 1,    43,      90,      90,      90,      90, ...
  0, 1,   352,    1879,    2520,    2520,    2520, ...
  0, 1,  3114,   47024,  102011,  113400,  113400, ...
  0, 1, 29004, 1331664, 5176504, 7235651, 7484400, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened
Ferenc Balogh, A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length, arXiv:1505.01389, 2015
Shalosh B. Ekhad and Doron Zeilberger, The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r, 2014

Columns k=0-4 give: A000007, A000012, A220097, A266734, A266735.
Main diagonal gives A000680.
First lower diagonal gives A267532.
Cf. A214015, A267480.

A(n,k) = Sum_{i=0..k} A267480(n,i).

A060538 Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 20, 90, 24, 1, 70, 1680, 2520, 120, 1, 252, 34650, 369600, 113400, 720, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1, 3432, 17153136, 11732745024, 305540235000, 137225088000, 681080400, 40320, 1, 12870

Offset: 1

Henry Bottomley, Apr 02 2001

			       1        1        1        1
       2        6       20       70
       6       90     1680    34650
      24     2520   369600 63063000

Harry J. Smith, Table of n, a(n) for n = 1..210

Subtable of A187783.
Rows include A000012, A000984, A006480, A008977, A008978 etc.
Columns include A000142, A000680, A014606, A014608, A014609 etc.
Main diagonal is A034841.

T(n,k)=(n*k)!/k!^n;
for(n=1, 6, for(k=1, 6, print1(T(n,k), ", ")); print) \\ Harry J. Smith, Jul 06 2009

T(n, k) = (nk)!/k!^n = T(n-1, k)*binomial(nk, k) = T(n-1, k)*A060539(n, k) = A060540(n, k)*A000142(k).

A065040 Triangle read by rows: T(m,k) = exponent of the highest power of 2 dividing the binomial coefficient binomial(m,k).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 3, 1, 3, 2, 3, 0, 0, 0, 2, 2, 1, 1, 2, 2, 0, 0, 0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0

Offset: 0

Claude Lenormand (hlne.lenormand(AT)voono.net), Nov 05 2001

T(m,k) is the number of 'carries' that occur when adding k and m-k in base 2 using the traditional addition algorithm. - Tom Edgar, Jun 10 2014

			Triangle begins:
[0]
[0, 0]
[0, 1, 0]
[0, 0, 0, 0]
[0, 2, 1, 2, 0]
[0, 0, 1, 1, 0, 0]
[0, 1, 0, 2, 0, 1, 0]
[0, 0, 0, 0, 0, 0, 0, 0]
[0, 3, 2, 3, 1, 3, 2, 3, 0]
[0, 0, 2, 2, 1, 1, 2, 2, 0, 0]
[0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0]
... - _N. J. A. Sloane_, Aug 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..10010; the first 141 antidiagonals, flattened
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Row sums: A187059.

Cf. A007318, A007814, A001511, A000120, A047999, A049606, A000680, A048881, A011371, A005187, A000265, A001316, A001317, A243759.

A065040 := (n, k) -> padic[ordp](binomial(n, k), 2):
seq(seq(A065040(n,k), k=0..n), n=0..13); # Peter Luschny, Aug 15 2017

T[m_, k_] := IntegerExponent[Binomial[m, k], 2]; Table[T[m, k], {m, 0, 13}, {k, 0, m}] // Flatten (* Jean-François Alcover, Oct 06 2016 *)

T(m,k)=hammingweight(k)+hammingweight(m-k)-hammingweight(m)
for(m=0,9,for(k=0,m,print1(T(m,k)", "))) \\ Charles R Greathouse IV, Mar 26 2013

As an array f(i,j) = f(j,i) = T(i+j,j) read by antidiagonals: f(0,j) = 0, f(1,j) = A007814(j+1), f(i,j) = Sum_{k=0..i-1} (f(1,j+k) - f(1,k)). [corrected by Kevin Ryde, Oct 07 2021]
The n-th term a(n) is equal to the binomial coefficient binomial(m,k), where m = floor((1+sqrt(8*n+1))/2) - 1 and k = n - m(m+1)/2. Also a(n) = g(m) - g(k) - g(m-k), where g(x) = Sum_{i=1..floor(log_2(x))} floor(x/2^i), m = floor((1+sqrt(8*n+1))/2) - 1, k = n - m(m+1)/2. - Hieronymus Fischer, May 05 2007
T(m,k) <= log_2 m, for m > 0. - Charles R Greathouse IV, Mar 26 2013
T(m,k) = log_2(A082907(m,k)). - Tom Edgar, Jun 10 2014
From Antti Karttunen, Oct 28 2014: (Start)
a(n) = A007814(A007318(n)).
a(n) * A047999(n) = 0 and a(n) + A047999(n) > 0 for all n.
(End)

Name clarified by Antti Karttunen, Oct 28 2014

A097591 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of odd length.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 6, 0, 17, 0, 1, 0, 70, 0, 49, 0, 1, 90, 0, 500, 0, 129, 0, 1, 0, 1890, 0, 2828, 0, 321, 0, 1, 2520, 0, 23100, 0, 13930, 0, 769, 0, 1, 0, 83160, 0, 215292, 0, 62634, 0, 1793, 0, 1, 113400, 0, 1549800, 0, 1697430, 0, 264072, 0, 4097, 0, 1

Offset: 0

Emeric Deutsch, Aug 29 2004

			Triangle starts:
     1;
     0,    1;
     1,    0,     1;
     0,    5,     0,    1;
     6,    0,    17,    0,     1;
     0,   70,     0,   49,     0,   1;
    90,    0,   500,    0,   129,   0,   1;
     0, 1890,     0, 2828,     0, 321,   0, 1;
  2520,    0, 23100,    0, 13930,   0, 769, 0, 1;
  ...
Row n has n+1 entries.
Example: T(3,1) = 5 because we have (123), 13(2), (2)13, 23(1) and (3)12 (the runs of odd length are shown between parentheses).

Alois P. Heinz, Rows n = 0..140, flattened

Bisections of columns k=0-1 give: A000680, A302910.
Row sums give A000142.
T(n+1,n-1) gives A000337.
T(4n,2n) gives A308962.
Cf. A096654, A097592.

G:=t^2/(1-t*x-(1-t^2)*exp(-t*x)): Gser:=simplify(series(G,x=0,12)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(expand(n!*coeff(Gser,x^n))) od: seq(seq(coeff(t*P[n],t^k),k=1..n+1),n=0..11);
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, x^t, expand(
      add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+
      add(b(u-j, o+j-1, 1)*x^t, j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 19 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, x^t, Expand[Sum[b[u+j-1, o-j, Mod[t+1, 2]], {j, 1, o}] + Sum[b[u-j, o+j-1, 1]*x^t, {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

E.g.f.: t^2/[1-tx-(1-t^2)exp(-tx)].

Sum_{k=1..n} k * T(n,k) = A096654(n-1) for n > 0. - Alois P. Heinz, Jul 03 2019

A268437 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](1/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 4, 6, 0, 30, 120, 90, 0, 336, 2800, 5040, 2520, 0, 5040, 80640, 264600, 302400, 113400, 0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400, 0, 2162160, 118198080, 983782800, 2986663680, 4162158000, 2724321600, 681080400

Offset: 0

Peter Luschny, Mar 07 2016

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

			[1],
[0, 1],
[0, 4, 6],
[0, 30, 120, 90],
[0, 336, 2800, 5040, 2520],
[0, 5040, 80640, 264600, 302400, 113400],
[0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400].

Peter Luschny, The P-transform.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Cf. A000680, A001761, A268438, A269939, A269941.

A268437 := proc(n,k) local F,T;
  F := proc(n,k) option remember;
  `if`(n=0 and k=0, 1,`if`(n=k, (4*n-2)*F(n-1,k-1),
  F(n-1,k)*(n+k))) end;
  T := proc(n,k) option remember;
  `if`(k=0 and n=0, 1,`if`(k<=0 or k>n, 0,
  (4*n-2)*n*(k*T(n-1,k)+(n+k-1)*T(n-1,k-1)))) end;
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268437(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268437_row := n -> PTrans(n, n->1/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(print(A268437_row(n)),n=0..8);

T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] StirlingS2[n+m, m], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)

A268437 = lambda n, k: (factorial(2*n)/falling_factorial(n+k, n))*sum((-1)^(m+k)* binomial(n+k, n+m)*stirling_number2(n+m, m) for m in (0..k))
for n in (0..7): print([A268437(n, m) for m in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(n+1), lambda n, k: (-1)^k* factorial(2*n))

T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling2(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269939(n,k).
T(n,1) = (2*n)!/(n+1)! = A001761(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.

A268438 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 8, 6, 0, 180, 240, 90, 0, 8064, 14560, 10080, 2520, 0, 604800, 1330560, 1285200, 604800, 113400, 0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400, 0, 10897286400, 30858347520, 43770767040, 36970053120, 18918900000, 5448643200, 681080400

Offset: 0

Peter Luschny, Mar 07 2016

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

			Triangle starts:
[1],
[0, 1],
[0, 8,        6],
[0, 180,      240,       90],
[0, 8064,     14560,     10080,     2520],
[0, 604800,   1330560,   1285200,   604800,    113400],
[0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400].

Peter Luschny, The P-transform.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Cf. A000680, A060593, A268437, A269940, A269941.

A268438 := proc(n,k) local F,T;
  F := proc(n,k) option remember;
  `if`(n=0 and k=0, 1,`if`(n=k, (4*n-2)*F(n-1,k-1),
  F(n-1,k)*(n+k))) end;
  T := proc(n, k) option remember;
  `if`(k=0 and n=0, 1,`if`(k<=0 or k>n, 0,
  (4*n-2)*n*(n+k-1)*(T(n-1,k)+T(n-1,k-1)))) end:
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268438(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268438_row := n -> PTrans(n, n->n/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(lprint(A268438_row(n)), n=0..8);

T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)

A268438 = lambda n,k: (factorial(2*n)/falling_factorial(n+k,n))*sum((-1)^(m+k)* binomial(n+k,n+m)*stirling_number1(n+m,m) for m in (0..k))
for n in (0..7): print([A268438(n,m) for m in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n))

T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling1(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269940(n,k).
T(n,1) = (2*n)!/(n+1) = A060593(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.

cp's OEIS Frontend

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A154283 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(2n+1,i) binomial(k+2-i,2)^n, 0 <= k <= 2*(n-1).

A268437 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](1/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

A268438 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.