A129505 -id:A129505 - cp's OEIS frontend

A238685 a(n) = n! * A129505(n) * (-1)^(n+1).

1, -6, 210, -17640, 2693880, -649479600, 226750764240, -108116216208000, 67478689070432640, -53382381970299782400, 52192613508738839136000, -61794396463636399635072000, 87121906773549083421777792000, -144222462676882552982237906688000

Offset: 1

Reinhard Zumkeller, Mar 02 2014

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..200

Cf. A000142, A008275, A048594, A129505.

a238685 n = a000142 n * a008275 (2 * n - 1) n

Array[#!*StirlingS1[2 # - 1, #] &, 14] (* Michael De Vlieger, Jan 24 2022 *)

a(n) = n!*stirling(2*n-1,n, 1); \\ Michel Marcus, Jan 24 2022

from math import factorial
from sympy.functions.combinatorial.numbers import stirling
def A238685(n): return factorial(n)*stirling((n<<1)-1,n,kind=1,signed=True) # Chai Wah Wu, Jun 09 2025

a(n) = A000142(n) * A008275(2*n-1,n).

a(n) = A048594(2*n-1,n).

A130534 Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 11, 6, 1, 24, 50, 35, 10, 1, 120, 274, 225, 85, 15, 1, 720, 1764, 1624, 735, 175, 21, 1, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1

Offset: 0

Philippe Deléham, Aug 09 2007

This triangle is an unsigned version of the triangle of Stirling numbers of the first kind, A008275, which is the main entry for these numbers. - N. J. A. Sloane, Jan 25 2011
Or, triangle T(n,k), 0 <= k <= n, read by rows given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.
Reversal of A094638.
Equals A132393*A007318, as infinite lower triangular matrices. - Philippe Deléham, Nov 13 2007
From Johannes W. Meijer, Oct 07 2009: (Start)
The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the exponential integrals E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + n*(n+1)/x^2 - n*(n+1)*(n+2)/x^3 + ...), see Abramowitz and Stegun. This formula follows from the general formula for the asymptotic expansion, see A163932. We rewrite E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + (n^2+n)/x^2 - (2*n+3*n^2+n^3)/x^3 + (6*n+11*n^2+6*n^3+n^4)/x^3 - ...) and observe that the T(n,m) are the polynomials coefficients in the denominators. Looking at the a(n,m) formula of A028421, A163932 and A163934, and shifting the offset given above to 1, we can write T(n-1,m-1) = a(n,m) = (-1)^(n+m)*Stirling1(n,m), see the Maple program.
The asymptotic expansion leads for values of n from one to eleven to known sequences, see the cross-references. With these sequences one can form the triangles A008279 (right-hand columns) and A094587 (left-hand columns).
See A163936 for information about the o.g.f.s. of the right-hand columns of this triangle.
(End)
The number of elements greater than i to the left of i in a permutation gives the i-th element of the inversion vector. (Skiena-Pemmaraju 2003, p. 69.) T(n,k) is the number of n-permutations that have exactly k 0's in their inversion vector. See evidence in Mathematica code below. - Geoffrey Critzer, May 07 2010
T(n,k) counts the rooted trees with k+1 trunks in forests of "naturally grown" rooted trees with n+2 nodes. This corresponds to sums of coefficients of iterated derivatives representing vectors, Lie derivatives, or infinitesimal generators for flow fields and formal group laws. Cf. links in A139605. - Tom Copeland, Mar 23 2014
A refinement is A036039. - Tom Copeland, Mar 30 2014
From Tom Copeland, Apr 05 2014: (Start)
With initial n=1 and row polynomials of T as p(n,x)=x(x+1)...(x+n-1), the powers of x correspond to the number of trunks of the rooted trees of the "naturally-grown" forest referred to above. With each trunk allowed m colors, p(n,m) gives the number of such non-plane colored trees for the forest with each tree having n+1 vertices.
p(2,m) = m + m^2 = A002378(m) = 2*A000217(m) = 2*(first subdiag of |A238363|).
p(3,m) = 2m + 3m^2 + m^3 = A007531(m+2) = 3*A007290(m+2) = 3*(second subdiag A238363).
p(4,m) = 6m + 11m^2 + 6m^3 + m^4 = A052762(m+3) = 4*A033487(m) = 4*(third subdiag).
From the Joni et al. link, p(n,m) also represents the disposition of n distinguishable flags on m distinguishable flagpoles.
The chromatic polynomial for the complete graph K_n is the falling factorial, which encodes the colorings of the n vertices of K_n and gives a shifted version of p(n,m).
E.g.f. for the row polynomials: (1-y)^(-x).
(End)
A relation to derivatives of the determinant |V(n)| of the n X n Vandermonde matrix V(n) in the indeterminates c(1) thru c(n):
  |V(n)| = Product_{1<=jTom Copeland, Apr 10 2014
From Peter Bala, Jul 21 2014: (Start)
Let M denote the lower unit triangular array A094587 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
  /I_k 0\
  \ 0  M/
having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well defined). See the Example section. (End)
For the relation of this rising factorial to the moments of Viennot's Laguerre stories, see the Hetyei link, p. 4. - Tom Copeland, Oct 01 2015
Can also be seen as the Bell transform of n! without column 0 (and shifted enumeration). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle  T(n,k) begins:
n\k         0        1        2       3       4      5      6     7    8  9 10
n=0:        1
n=1:        1        1
n=2:        2        3        1
n=3:        6       11        6       1
n=4:       24       50       35      10       1
n=5:      120      274      225      85      15      1
n=6:      720     1764     1624     735     175     21      1
n=7:     5040    13068    13132    6769    1960    322     28     1
n=8:    40320   109584   118124   67284   22449   4536    546    36    1
n=9:   362880  1026576  1172700  723680  269325  63273   9450   870   45  1
n=10: 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55  1
[Reformatted and extended by _Wolfdieter Lang_, Feb 05 2013]
T(3,2) = 6 because there are 6 permutations of {1,2,3,4} that have exactly 2 0's in their inversion vector: {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2}, {2, 1, 3, 4},{2, 3, 1, 4}, {2, 3, 4, 1}. The respective inversion vectors are {0, 0, 1}, {0, 1, 0}, {0, 2, 0}, {1, 0, 0}, {2, 0, 0}, {3, 0, 0}. - _Geoffrey Critzer_, May 07 2010
T(3,1)=11 since there are exactly 11 permutations of {1,2,3,4} with exactly 2 cycles, namely, (1)(234), (1)(243), (2)(134), (2)(143), (3)(124), (3)(142), (4)(123), (4)(143), (12)(34), (13)(24), and (14)(23). - _Dennis P. Walsh_, Jan 25 2011
From _Peter Bala_, Jul 21 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
  / 1          \/1        \/1        \      / 1           \
  | 1  1       ||0 1      ||0 1      |      | 1  1        |
  | 2  2  1    ||0 1 1    ||0 0 1    |... = | 2  3  1     |
  | 6  6  3 1  ||0 2 2 1  ||0 0 1 1  |      | 6 11  6  1  |
  |24 24 12 4 1||0 6 6 3 1||0 0 2 2 1|      |24 50 35 10 1|
  |...         ||...      ||...      |      |...          |
(End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 93-94.
Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge University Press, 2003, pp. 69-71. [Geoffrey Critzer, May 07 2010]

T. D. Noe, Rows n = 0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251. [From _Johannes W. Meijer_, Oct 07 2009]
A. Chervov, Decomplexification of the Capelli identities and holomorphic factorization, arxiv 1203.5759 [math.QA], Mar 2012. [_Tom Copeland_, Apr 10 2014]
FindStat - Combinatorial Statistic Finder, The number of saliances of the permutation, The number of cycles in the cycle decomposition of a permutation.
Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009.
S. Joni, G. Rota, and B. Sagan, From Sets to Functions: Three Elementary Examples, Discrete Mathematics, vol. 37, no. 2-3, pp. 193-202, 1981. [_Tom Copeland_, Apr 05 2014]
Matthieu Josuat-Verges, A q-analog of Schläfli and Gould identities on Stirling numbers, Preprint, arXiv:1610.02965 [math.CO], 2016.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021.
Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.
Dennis Walsh, A short note on unsigned Stirling numbers

See A008275, which is the main entry for these numbers; A094638 (reversed rows).
Diagonals: A000012 A000217 A000914 A001303 A000915 A053567 A112002 A191685. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.
From Johannes W. Meijer, Oct 07 2009: (Start)
Row sums equal A000142.
The asymptotic expansions lead to A000142 (n=1), A000142(n=2; minus a(0)), A001710 (n=3), A001715 (n=4), A001720 (n=5), A001725 (n=6), A001730 (n=7), A049388 (n=8), A049389 (n=9), A049398 (n=10), A051431 (n=11), A008279 and A094587.
Cf. A163931 (E(x,m,n)), A028421 (m=2), A163932 (m=3), A163934 (m=4), A163936.
(End)
Cf. A136662.

a130534 n k = a130534_tabl !! n !! k
a130534_row n = a130534_tabl !! n
a130534_tabl = map (map abs) a008275_tabl
-- Reinhard Zumkeller, Mar 18 2013

with(combinat): A130534 := proc(n,m): (-1)^(n+m)*stirling1(n+1,m+1) end proc: seq(seq(A130534(n,m), m=0..n), n=0..10); # Johannes W. Meijer, Oct 07 2009, revised Sep 11 2012
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0 (and shifts the enumeration).
BellMatrix(n -> n!, 9); # Peter Luschny, Jan 27 2016

Table[Table[ Length[Select[Map[ToInversionVector, Permutations[m]], Count[ #, 0] == n &]], {n, 0, m - 1}], {m, 0, 8}] // Grid (* Geoffrey Critzer, May 07 2010 *)
rows = 10;
t = Range[0, rows]!;
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

T(0,0) = 1, T(n,k) = 0 if k > n or if n < 0, T(n,k) = T(n-1,k-1) + n*T(n-1,k). T(n,0) = n! = A000142(n). T(2*n,n) = A129505(n+1). Sum_{k=0..n} T(n,k) = (n+1)! = A000142(n+1). Sum_{k=0..n} T(n,k)^2 = A047796(n+1). T(n,k) = |Stirling1(n+1,k+1)|, see A008275. (x+1)(x+2)...(x+n) = Sum_{k=0..n} T(n,k)*x^k. [Corrected by Arie Bos, Jul 11 2008]
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, respectively. - Philippe Deléham, Nov 13 2007
For k=1..n, let A={a_1,a_2,...,a_k} denote a size-k subset of {1,2,...,n}. Then T(n,n-k) = Sum(Product_{i=1..k} a_i) where the sum is over all subsets A. For example, T(4,1)=50 since 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4 = 50. - Dennis P. Walsh, Jan 25 2011
The preceding formula means T(n,k) = sigma_{n-k}(1,2,3,..,n) with the (n-k)-th elementary symmetric function sigma with the indeterminates chosen as 1,2,...,n. See the Oct 24 2011 comment in A094638 with sigma called there a. - Wolfdieter Lang, Feb 06 2013
From Gary W. Adamson, Jul 08 2011: (Start)
n-th row of the triangle = top row of M^n, where M is the production matrix:
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  1, 4, 6, 4, 1;
  ... (End)
Exponential Riordan array [1/(1 - x), log(1/(1 - x))]. Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (n + 1)!/(n + 1 - i)!*T(n-i,k). - Peter Bala, Jul 21 2014

A187646 (Signless) Central Stirling numbers of the first kind s(2n,n).

Original entry on oeis.org

1, 1, 11, 225, 6769, 269325, 13339535, 790943153, 54631129553, 4308105301929, 381922055502195, 37600535086859745, 4070384057007569521, 480544558742733545125, 61445535102359115635655, 8459574446076318147830625, 1247677142707273537964543265, 196258640868140652967646352465

Offset: 0

Emanuele Munarini, Mar 12 2011

Number of permutations with n cycles on a set of size 2n.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A008275, A007820, A129505, A132393, A187235, A237993, A242676, A285862.

Maple
```
seq(abs(Stirling1(2*n,n)), n=0..20);
```

Table[Abs[StirlingS1[2n, n]], {n, 0, 12}]
N[1 + 1/(2 LambertW[-1, -Exp[-1/2]/2]), 50] (* The constant z in the asymptotic - Vladimir Reshetnikov, Oct 08 2016 *)

makelist(abs(stirling1(2*n,n)),n,0,12);

for(n=0,50, print1(abs(stirling(2*n, n, 1)), ", ")) \\ G. C. Greubel, Nov 09 2017

Asymptotic: a(n) ~ (2*n/(e*z*(1-z)))^n*sqrt((1-z)/(2*Pi*n*(2z-1))), where z=0.715331862959... is a root of the equation z = 2*(z-1)*log(1-z). - Vaclav Kotesovec, May 30 2011
Equivalent: a(n) ~ n!*(2*r^2/(r-1))^n/(2*Pi*n*sqrt(r-2)), where r=A226278. - Natalia L. Skirrow, Jul 13 2025
From Seiichi Manyama, May 20 2025: (Start)
a(n) = A132393(2*n,n).
a(n) = (2*n)! * [x^(2*n)] (-log(1 - x))^n / n!. (End)

A129506 Number of partitions of a {2n-1}-set into n nonempty subsets.

Original entry on oeis.org

1, 3, 25, 350, 6951, 179487, 5715424, 216627840, 9528822303, 477297033785, 26826851689001, 1672162773483930, 114485073343744260, 8541149231801585700, 689692892575539953400, 59932861644880019603520, 5576731051262006158950735, 553234633385550257808059085

Offset: 1

Paul D. Hanna, Apr 18 2007

nonn

B^{-1}(x) = Sum_{n>0} a(n)/(2*n-1)!*(n-1)! x^n is inverse function for x*B(x), where B(x) is g.f. for Bernoulli number (see A027641). - Vladimir Kruchinin, Jan 19 2012

			G.f.: A(x) = x + 3*x^2 + 25*x^3 + 350*x^4 + 6951*x^5 + 179487*x^6 + ... where A(x) = 1^1*x*exp(-1^2*x) + 2^3*exp(-2^2*x)*x^2/2! + 3^5*exp(-3^2*x)*x^3/3! + 4^7*exp(-4^2*x)*x^4/4! + 5^9*exp(-5^2*x)*x^5/5! + ... forms a power series in x with integer coefficients. - _Paul D. Hanna_, Oct 15 2012

Alois P. Heinz, Table of n, a(n) for n = 1..200
D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequences, Vol. 15 (2012), article 12.9.3.

Cf. A008277, A129505, A217900, A217910, A217913, A258170.

a:= n-> Stirling2(2*n-1, n):
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 15 2013

a[n_] := Sum[ Binomial[2*n - 2, j]*StirlingS2[2*n - j - 2, n-1], {j, 0, n-1}]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Jun 14 2013, after Vladimir Kruchinin *)
Table[StirlingS2[2*n-1,n], {n, 1, 20}] (* Vaclav Kotesovec, Dec 15 2013 *)

a(n):=((2*n-1)*((sum((stirling2(2*i-1, i)*binomial(2*n-2, 2*i-1)*stirling2(2*(n-i)-1, n-i-1))/((n-i-1)*binomial(n-1, i)), i, 1, n-2))+(n-1)* stirling2(2*n-3, n-1)+stirling2(2*n-2, n-1)))/(n);
  makelist(a(n),n,1,10); /* Vladimir Kruchinin, Feb 28 2013 */

a(n)=polcoeff(1/prod(k=1,n,1-k*x +x*O(x^n)),n-1)

vector(66, n, abs( stirling(2*n-1, n, 2) ) ) /* Joerg Arndt, Jun 09 2012 */

{a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^(2*n-1))} \\ Paul D. Hanna, Oct 15 2012

{a(n)=polcoeff(sum(m=1,n,m^(2*m-1)*x^m*exp(-m^2*x+x*O(x^n))/m!),n)}
for(n=1,20,print1(a(n),", "))

Central Stirling numbers of the second kind: a(n) = A008277(2n-1,n) for n >= 1.
G.f.: Sum_{n>=1} n^(2*n-1) * exp(-n^2*x) * x^n / n!, an integer series. - Paul D. Hanna, Oct 15 2012
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(2*n-1). - Paul D. Hanna, Oct 15 2012
a(n) = ((2*n-1)*((sum(i=1..n-2, (stirling2(2*i-1,i)*C(2*n-2,2*i-1)*stirling2(2*(n-i)-1,n-i-1))/((n-i-1)*C(n-1,i))))+(n-1)*stirling2(2*n-3,n-1) +stirling2(2*n-2,n-1)))/n. - Vladimir Kruchinin, Feb 28 2013
a(n-1) = sum(j=0..n, binomial(2*n,j)*stirling2(2*n-j,n)). - Vladimir Kruchinin, Jun 14 2013
a(n) ~ 2^(2*n-3/2) * n^(n-3/2) * (2-c)^(1-n) / (sqrt(Pi*(1-c)) * exp(n) * c^n), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - Vaclav Kotesovec, Dec 15 2013
a(n) = A258170(2*n-1,n). - Alois P. Heinz, Mar 16 2018

A234324 Central terms of the triangle of central factorial numbers (A008955).

Original entry on oeis.org

1, 5, 273, 44473, 14739153, 8261931405, 7026231453265, 8439654758970225, 13611213226804376865, 28383081191068916580565, 74326386672885754888959569, 238812235698229573137588936105, 923793013650701305781038113833585, 4235104161629281000608041716747118685

Offset: 0

Reinhard Zumkeller, Dec 24 2013

nonn

a(n) = A008955(2*n,n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..120

Cf. A008955, A129505.

Haskell
```
a234324 n = a008955 (2 * n) n
```

b:= proc(n, k) option remember; `if`(k=0, 1,
      add(b(j-1, k-1)*j^2, j=1..n))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 19 2022

Flatten[{1, Table[Coefficient[Expand[Product[1 + k^2*x, {k, 0, 2*n}]], x^n], {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 28 2017 *)
Table[Sum[(-1)^(n-j) * StirlingS1[2*n+1, 2*n+1-j] * StirlingS1[2*n+1, j+1], {j, 0, 2*n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 28 2017 *)

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 30.472614208903344955058219429421999299236787591960717... and c = 0.27436634881777520262458169083560234658... - Vaclav Kotesovec, Aug 28 2017

A348084 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k x).

Original entry on oeis.org

1, 3, 65, 2646, 159027, 12662650, 1256328866, 149304004500, 20677182465555, 3270191625210510, 581535955088511150, 114860168436414644100, 24947615188488584876910, 5909902157669174347277556, 1516413100622001261250104100, 418965782063742792530650053000

Offset: 0

Seiichi Manyama, Sep 28 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..300

Cf. A007820 (S2(2*n,n)), A008277 (S2(n,k)), A129505, A348085.

a(n) = polcoef(1/prod(k=1, 2*n, 1-k*x+x*O(x^n)), n);

PARI
```
a(n) = stirling(3*n, 2*n, 2);
```

a(n) = sum(k=0, 2*n, (-1)^k*k^(3*n)*binomial(2*n, k))/(2*n)!;

from sympy.functions.combinatorial.numbers import stirling
def A348084(n): return stirling(3*n,2*n) # Chai Wah Wu, Jun 09 2025

a(n) = S2(3*n,2*n), where S2(n,k) is Stirling numbers of the second kind.
a(n) = (1/(2*n)!) * Sum_{k=0..2*n} (-1)^k * k^(3*n) * binomial(2*n,k).
a(n) ~ 3^(3*n) * n^(n - 1/2) / (sqrt(Pi*(1-c)) * 2^(n+1) * exp(n) * (3 - 2*c)^n * c^(2*n)), where c = -LambertW(-3*exp(-3/2)/2) = 0.62578253420128292093838... - Vaclav Kotesovec, Oct 02 2021

A348024 a(n) = [x^n] Product_{k=1..2n} (x + (-1)^k k).

Original entry on oeis.org

1, 1, -13, -87, 2609, 34965, -1638841, -33411007, 2128380881, 58501278297, -4709853961605, -163166745402375, 15875893807564465, 664163782026582877, -75766955207796981345, -3716140360596415682175, 486265092408065178623265, 27363821966293458296896305, -4039473192629570940395992765

Offset: 0

Seiichi Manyama, Sep 25 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..350

Cf. A047991, A129505, A348023.

a(n) = polcoef(prod(k=1, 2*n, x+(-1)^k*k), n);

a(n) = A047991(2*n,n).

A226057 E.g.f. A(x) satisfies: A(x)^2 = -xlog(1-A(x)) where A(x) = Sum_{n>=1} a(n)x^n/n!^2.

Original entry on oeis.org

1, 2, 21, 504, 21380, 1405800, 132139140, 16801276800, 2775758497344, 577868994460800, 147973478687496000, 45703277816543424000, 16753246307626306832640, 7190163806348621417679360, 3571395525388698501285792000

Offset: 1

Paul D. Hanna, May 24 2013

nonn

Name is directly from a formula for A129505 given by Vladimir Kruchinin.

			E.g.f.: A(x) = x + 2*x^2/2!^2 + 21*x^3/3!^2 + 504*x^4/4!^2 + 21380*x^5/5!^2 +...
where
A(x)^2 = 2*x^2/2! + 6*x^3/3! + 34*x^4/4! + 280*x^5/5! + 3013*x^6/6! + 39963*x^7/7! + 629541*x^8/8! +...
and
-log(1-A(x)) = 2*x/2! + 6*x^2/3! + 34*x^3/4! + 280*x^4/5! + 3013*x^5/6! +...

D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

Cf. A204248, A129505.

{a(n)=polcoeff(prod(k=0, 2*n-2, 1+k*x), n-1)*n!^2*(n-1)!/(2*n-1)!}

from math import factorial, comb
from sympy.functions.combinatorial.numbers import stirling
def A226057(n): return factorial(n)*stirling(m:=(n<<1)-1,n,kind=1)//comb(m,n-1) # Chai Wah Wu, Jun 08 2025

a(n) = n!^2*(n-1)!/(2*n-1)! * {[x^(n-1)] Product_{k=0..2*n-2} (1+k*x)}.
a(n) = n!^2*(n-1)!/(2*n-1)! * A129505(n), where A129505(n) = number of permutations of 2n-1 objects with exactly n cycles.
a(n) = n*A204248(n-1), where A204248(n) = permanent of the n-th principal submatrix of A002024.

cp's OEIS Frontend

A238685 a(n) = n! * A129505(n) * (-1)^(n+1).

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A130534 Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

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A187646 (Signless) Central Stirling numbers of the first kind s(2n,n).

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A129506 Number of partitions of a {2n-1}-set into n nonempty subsets.

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A234324 Central terms of the triangle of central factorial numbers (A008955).

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A348084 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - k * x).

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A348084 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k x).

A348024 a(n) = [x^n] Product_{k=1..2n} (x + (-1)^k k).

A226057 E.g.f. A(x) satisfies: A(x)^2 = -xlog(1-A(x)) where A(x) = Sum_{n>=1} a(n)x^n/n!^2.