A001044 -id:A001044 - cp's OEIS frontend

A007407 a(n) = denominator of Sum_{k=1..n} 1/k^2.

1, 4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 10838475198270720, 221193371393280

Offset: 1

N. J. A. Sloane, Mira Bernstein

Denominators of the Eulerian numbers T(-2,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - Michael J. Collins, Oct 10 2024

			1/1^2 + 1/2^2 + 1/3^2 = 1/1 + 1/4 + 1/9 = 49/36, so a(3) = 36. - _Jon E. Schoenfield_, Dec 26 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..1152 (terms 1..200 from T. D. Noe)
D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.

Cf. A007406 (numerators), A000290, A035166.

import Data.Ratio ((%), denominator)
a007407 n = a007407_list !! (n-1)
a007407_list = map denominator $
                   scanl1 (+) $ map (1 %) $ tail a000290_list
-- Reinhard Zumkeller, Jul 06 2012

ZL:=n->sum(1/i^2, i=2..n): a:=n->floor(denom(ZL(n))): seq(a(n), n=1..21); # Zerinvary Lajos, Mar 28 2007

s=0;lst={};Do[s+=n^2/n^4;AppendTo[lst,Denominator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[Denominator[Pi^2/6 - Zeta[2, x]], {x, 1, 22}] (* Artur Jasinski, Mar 03 2010 *)
Denominator[Accumulate[1/Range[30]^2]] (* Harvey P. Dale, Nov 08 2012 *)

a(n)=denominator(sum(k=1,n,1/k^2)) \\ Charles R Greathouse IV, Nov 20 2012

from fractions import Fraction
def A007407(n): return sum(Fraction(1,k**2) for k in range(1,n+1)).denominator # Chai Wah Wu, Apr 03 2021

a(n) = denominator of (Pi^2)/6 - zeta(2, x). - Artur Jasinski, Mar 03 2010

a(n) = A001044(n) / gcd(A001819(n), A001044(n)). - Daniel Suteu, Dec 25 2016

A008955 Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 4, 1, 14, 49, 36, 1, 30, 273, 820, 576, 1, 55, 1023, 7645, 21076, 14400, 1, 91, 3003, 44473, 296296, 773136, 518400, 1, 140, 7462, 191620, 2475473, 15291640, 38402064, 25401600, 1, 204, 16422, 669188, 14739153, 173721912, 1017067024, 2483133696, 1625702400

Offset: 0

N. J. A. Sloane

Discussion of Central Factorial Numbers by N. J. A. Sloane, Feb 01 2011: (Start)
Here is Riordan's definition of the central factorial numbers t(n,k) given in Combinatorial Identities, Section 6.5:
For n >= 0, expand the polynomial
x^[n] = x*Product{i=1..n-1} (x+n/2-i) = Sum_{k=0..n} t(n,k)*x^k.
The t(n,k) are not always integers. The cases n even and n odd are best handled separately.
For n=2m, we have:
x^[2m] = Product_{i=0..m-1} (x^2-i^2) = Sum_{k=1..m} t(2m,2k)*x^(2k).
E.g. x^[8] = x^2(x^2-1^2)(x^2-2^2)(x^2-3^2) = x^8-14x^6+49x^4-36x^2,
which corresponds to row 4 of the present triangle.
So the m-th row of the present triangle gives the absolute values of the coefficients in the expansion of Product_{i=0..m-1} (x^2-i^2).
Equivalently, and simpler, the n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first.
For n odd, n=2m+1, we have:
x^[2m+1] = x*Product_{i=0..m-1}(x^2-((2i+1)/2)^2) = Sum_{k=0..m} t(2m+1,2k+1)*x^(2k+1).
E.g. x^[5] = x(x^2-(1/2)^2)(x^2-(3/2)^2) = x^5-10x^3/4+9x/16,
which corresponds to row 2 of the triangle in A008956.
We now rescale to get integers by replacing x by x/2 and multiplying by 2^(2m+1) (getting 1, -10, 9 from the example).
The result is that row m of triangle A008956 gives the coefficients in the expansion of x*Product_{i=0..m} (x^2-(2i+1)^2).
Equivalently, and simpler, the n-th row of A008956 gives the coefficients in the expansion of Product_{i=0..n-1} (x+(2i+1)^2), highest powers first.
Note that the n-th row of A182867 gives the coefficients in the expansion of Product_{i=1..n} (x+(2i)^2), highest powers first.
(End)
Contribution from Johannes W. Meijer, Jun 18 2009: (Start)
We define Beta(n-z,n+z)/Beta(n,n) = Gamma(n-z)*Gamma(n+z)/Gamma(n)^2 = sum(EG2[2m,n]*z^(2m), m = 0..infinity) with Beta(z,w) the Beta function. The EG2[2m,n] coefficients are quite interesting, see A161739. Our definition leads to EG2[2m,1] = 2*eta(2m) and the recurrence relation EG2[2m,n] = EG2[2m,n-1] - EG2[2m-2,n-1]/(n-1)^2 for m = -2, -1, 0, 1, 2, ... and n = 2, 3, ... , with eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. We found for the matrix coefficients EG2[2m,n] = sum((-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2,k=1..n) with the central factorial numbers t1(n,m) as defined above, see also the Maple program.
From the EG2 matrix we arrive at the ZG2 matrix, see A161739 for its odd counterpart, which is defined by ZG2[2m,1] = 2*zeta(2m) and the recurrence relation ZG2[2m,n] = ZG2[2m-2,n-1]/(n*(n-1))-(n-1)*ZG2[2m,n-1]/n for m = -2, -1, 0, 1, 2, ... and n = 2, 3, ... . We found for the ZG2[2m,n] = Sum_{k=1..n} (-1)^(k+1)*t1(n-1,k-1)* 2* zeta(2*m-2*n+2*k)/((n-1)!*(n)!), and we see that the central factorial numbers t1(n,m) once again play a crucial role.
(End)

			Triangle begins:
  1;
  1,   1;
  1,   5,   4;
  1,  14,  49,  36;
  1,  30, 273, 820, 576;
  ...

B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Alois P. Heinz, Rows n = 0..100, flattened (first 51 rows from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
R. H. Boels, Three particle superstring amplitudes with massive legs, arXiv preprint arXiv:1201.2655 [hep-th], 2012.
R. H. Boels and T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt, Central Factorial Numbers: Their main properties and some applications, Numerical Functional Analysis and Optimization, 10 (5&6), 419-488 (1989).
M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
T. L. Curtright and T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arxiv:1408.0767 [math-ph], 2014.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
Toshiki Matsusaka, Applications of Faà di Bruno's formula to partition traces, arXiv:2507.00404 [math.NT], 2025. See p. 5.
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
S. Shadrin, L. Spitz, and D. Zvonkine, On double Hurwitz numbers with completed cycles, J. Lond. Math. Soc., II. Ser. 86, No. 2, 407-432 (2012), Corollary 7.5.

Cf. A036969.
Columns include A000330, A000596, A000597. Right-hand columns include A001044, A001819, A001820, A001821. Row sums are in A101686.
Appears in A160464 (Eta triangle), A160474 (Zeta triangle), A160479 (ZL(n)), A161739 (RSEG2 triangle), A161742, A161743, A002195, A002196, A162440 (EG1 matrix), A162446 (ZG1 matrix) and A163927. - Johannes W. Meijer, Jun 18 2009, Jul 06 2009 and Aug 17 2009
Cf. A234324 (central terms).

T:= function(n,k)
    if k=0 then return 1;
    elif k=n then return (Factorial(n))^2;
    else return n^2*T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..8], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Sep 14 2019

a008955 n k = a008955_tabl !! n !! k
a008955_row n = a008955_tabl !! n
a008955_tabl = [1] : f [1] 1 1 where
   f xs u t = ys : f ys v (t * v) where
     ys = zipWith (+) (xs ++ [t^2]) ([0] ++ map (* u^2) (init xs) ++ [0])
     v = u + 1
-- Reinhard Zumkeller, Dec 24 2013

T:= func< n,k | Factorial(2*(n+1))*(&+[(-1)^j*Binomial(n,k-j)*(&+[2^(m-2*k)*StirlingFirst(2*(n-k+1)+m, 2*(n-k+1))*Binomial(2*(n-k+1)+2*j-1, 2*(n-k+1)+m-1)/Factorial(2*(n-k+1)+m): m in [0..2*j]]): j in [0..k]]) >;
[T(n,k): k in [0..n], n in [0..8]]; // G. C. Greubel, Sep 14 2019

nmax:=7: for n from 0 to nmax do t1(n, 0):=1: t1(n, n):=(n!)^2 end do: for n from 1 to nmax do for k from 1 to n-1 do t1(n, k) := t1(n-1, k-1)*n^2 + t1(n-1, k) end do: end do: seq(seq(t1(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 18 2009, Revised Sep 16 2012
t1 := proc(n,k)
        sum((-1)^j*stirling1(n+1,n+1-k+j)*stirling1(n+1,n+1-k-j),j=-k..k) ;
end proc: # Mircea Merca, Apr 02 2012
# third Maple program:
T:= proc(n, k) option remember; `if`(k=0, 1,
      add(T(j-1, k-1)*j^2, j=1..n))
    end:
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Feb 19 2022

t[n_, 0]=1; t[n_, n_]=(n!)^2; t[n_ , k_ ]:=t[n, k] = n^2*t[n-1, k-1] + t[n-1, k]; Flatten[Table[t[n, k], {n,0,8}, {k,0,n}] ][[1 ;; 42]]
(* Jean-François Alcover, May 30 2011, after recurrence formula *)

T(n,m):=(2*(n+1))!*sum((-1)^k*binomial(n,m-k)*sum((2^(i-2*m)*stirling1(2*(n-m+1)+i,2*(n-m+1))*binomial(2*(n-m+1)+2*k-1,2*(n-m+1)+i-1))/(2*(n-m+1)+i)!,i,0,2*k),k,0,m); /* Vladimir Kruchinin, Oct 05 2013 */

T(n,k)=if(k==0,1, if(k==n, (n!)^2, n^2*T(n-1, k-1) + T(n-1, k)));
for(n=0,8, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 14 2019

# This triangle is (0,0)-based.
def A008955(n, k) :
    if k==0 : return 1
    if k==n : return factorial(n)^2
    return n^2*A008955(n-1, k-1) + A008955(n-1, k)
for n in (0..7) : print([A008955(n, k) for k in (0..n)]) # Peter Luschny, Feb 04 2012

The n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first (see Comments section).
The triangle can be obtained from the recurrence t1(n,k) = n^2*t1(n-1,k-1) + t1(n-1,k) with t1(n,0) = 1 and t1(n,n) = (n!)^2.
t1(n,k) = Sum_{j=-k..k} (-1)^j*s(n+1,n+1-k+j)*s(n+1,n+1-k-j) = Sum_{j=0..2*(n+1-k)} (-1)^(n+1-k+j)*s(n+1,j)*s(n+1,2*(n+1-k)-j), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 02 2012
E.g.f.: cosh(2/sqrt(t)*asin(sqrt(t)*z/2)) = 1 + z^2/2! + (1 + t)*z^4/4! + (1 + 5*t + 4*t^2)*z^6/6! + ... (see Berndt, p.263 and p.306). - Peter Bala, Aug 29 2012
T(n,m) = (2*(n+1))!*Sum_{k=0..m} ((-1)^k*binomial(n,m-k)*Sum_{i=0..2*k} ((2^(i-2*m)*stirling1(2*(n-m+1)+i,2*(n-m+1))*binomial(2*(n-m+1)+2*k-1, 2*(n-m+1)+i-1))/(2*(n-m+1)+i)!)). - Vladimir Kruchinin, Oct 05 2013

There's an error in the last column of Riordan's table (change 46076 to 21076).
More terms from Vladeta Jovovic, Apr 16 2000
Link added and cross-references edited by Johannes W. Meijer, Aug 17 2009
Discussion of Riordan's definition of central factorial numbers added by N. J. A. Sloane, Feb 01 2011

A088311 Number of sets of lists with distinct list sizes, cf. A000262.

Original entry on oeis.org

1, 1, 2, 12, 48, 360, 2880, 25200, 241920, 2903040, 36288000, 479001600, 7185024000, 112086374400, 1917922406400, 35307207936000, 669529276416000, 13516122267648000, 294509190463488000, 6568835422076928000, 155705728523304960000, 3882911605049917440000

Offset: 0

Vladeta Jovovic, Nov 05 2003

nonn

a(n) also enumerates ordered pairs of permutation functions on n elements where f(g(x)) = g(g(f(x))). - Chad Brewbaker, Mar 27 2014

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

Cf. A000009, A007837, A007838.

Other ordered permutation function pair relations are A000012, A000085, A000142, A001044, A053529.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( (&*[1+x^j: j in [1..m+2]]) ))); // G. C. Greubel, Dec 14 2022

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> n!*b(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nn = 19; Drop[ Range[0, nn]! CoefficientList[ Series[ Product[1 + x^i, {i,nn}], {x,0,nn}], x], 0] (* Geoffrey Critzer, Aug 05 2013; adapted to new offset by Vincenzo Librandi, Mar 28 2014 *)
nmax = 20; CoefficientList[Series[Product[1/(1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 19 2015 *)

my(x='x+O('x^66)); Vec(serlaplace(eta(x^2)/eta(x))) \\ Joerg Arndt, Aug 06 2013

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1) #  Peter Luschny's code of A000009 and A166861
b = EulerTransform(a)
[factorial(n)*b(n) for n in range(41)] # G. C. Greubel, Dec 14 2022

E.g.f: Product_{m>0} (1+x^m).

a(n) = n! * A000009(n).

Prepended a(0) = 1, Joerg Arndt, Aug 06 2013

A001819 Central factorial numbers: second right-hand column of triangle A008955.

Original entry on oeis.org

0, 1, 5, 49, 820, 21076, 773136, 38402064, 2483133696, 202759531776, 20407635072000, 2482492033152000, 359072203696128000, 60912644957448192000, 11977654199703478272000, 2702572249389834608640000

Offset: 0

N. J. A. Sloane

Coefficient of x^2 in Product_{k=0..n}(x + k^2). - Ralf Stephan, Aug 22 2004
p divides a(p-1) for prime p > 3. p divides a((p-1)/2) for prime p > 3. For prime p, p^2 divides a(n) for n > 2*p+1. - Alexander Adamchuk, Jul 11 2006; last comment corrected by Michel Marcus, May 20 2020
The ratio a(n)/A001044(n) is the partial sum of the reciprocals of squares. E.g., a(4)/A001044(4) = 820/576 = 1/1 + 1/4 + 1/9 + 1/16. - Pierre CAMI, Oct 30 2006
a(n) is the (n-1)-st elementary symmetric function of the squares of the first n numbers. - Anton Zakharov, Nov 06 2016
Primes p such that p^2 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..253 (terms 0..50 from T. D. Noe)
Takao Komatsu, Convolution identities of poly-Cauchy numbers with level 2, arXiv:2003.12926 [math.NT], 2020.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
Index entries for sequences related to factorial numbers

Cf. A000254, A001044, A002455, A007406, A007407, A066989, A024167, A142995.
Second right-hand column of triangle A008955.
Equals row sums of A162990(n)/(n+1)^2 for n >= 1.

Table[Sum[1/i^2,{i,1,n}]/Product[1/i^2,{i,1,n}],{n,1,40}] (* Alexander Adamchuk, Jul 11 2006 *)
Table[n!^2*HarmonicNumber[n, 2], {n, 0, 15}] (* Jean-François Alcover, May 09 2012, after Joe Keane *)

a(n)=n!^2*sum(k=1,n,1/k^2) \\ Charles R Greathouse IV, Nov 06 2016

a_n = (n!)^2 * Sum_{k=1..n} 1/k^2. - Joe Keane (jgk(AT)jgk.org)
a(n) ~ (1/3)*Pi^3*n*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
Sum_{n>=0} a(n)*x^n/n!^2 = polylog(2, x)/(1-x). - Vladeta Jovovic, Jan 23 2003
a(n) = Sum_{i=1..n} 1/i^2 / Product_{i=1..n} 1/i^2. - Alexander Adamchuk, Jul 11 2006
a(0) = 0, a(n) = a(n-1)*n^2 + A001044(n-1). E.g., a(1) = 0*1 + 1 = 1 since A001044(0) = 1; a(2) = 1*2^2 + 1 = 5 since A001044(1) = 1; a(3) = 5*3^2 + 4 = 49 since A001044(2) = 4; and so on. - Pierre CAMI, Oct 30 2006
Recurrence: a(0) = 0, a(1) = 1, a(n+1) = (2*n^2 + 2*n + 1)*a(n) - n^4*a(n-1). The sequence b(n) = n!^2 satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(1 - 1^4/(5 - 2^4/(13 - 3^4/(25 - ... -(n-1)^4/((2*n^2 - 2*n + 1)))))), leading to the infinite continued fraction expansion zeta(2) = 1/(1-1^4/(5 - 2^4/(13 - 3^4/(25 - ... - n^4/((2*n^2 + 2*n + 1) - ...))))). Compare with A142995. Compare also with A024167 and A066989. - Peter Bala, Jul 18 2008
a(n)/(n!)^2 -> zeta(2) = A013661 as n -> infinity, rewriting the Keane formula. - Najam Haq (njmalhq(AT)yahoo.com), Jan 13 2010
a(n) = s(n+1,2)^2 - 2*s(n+1,1)*s(n+1,3), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 03 2012

Minor edits by Vaclav Kotesovec, Jan 28 2015

A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.

Original entry on oeis.org

1, 3, 45, 1575, 99225, 9823275, 1404728325, 273922023375, 69850115960625, 22561587455281875, 9002073394657468125, 4348001449619557104375, 2500100833531245335015625, 1687568062633590601135546875, 1321365793042101440689133203125

Offset: 0

Benoit Cloitre, Jan 17 2003

nonn

a(n) is the determinant of M(2n+1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k m(i,j)=i otherwise. - Adapted to offset 0, Rainer Rosenthal, Jun 19 2024
In the following two comments on the calculation of the terms using permanents, offset 1 is assumed. In the corresponding PARI code, this is implemented with offset 0. - Hugo Pfoertner, Jun 23 2024
(-1)^n*a(n)/2^(2n-1) is the permanent of the (m X m) matrix {1/(x_i-y_j), 1<=i<=m, 1<=j<=m}, where x_1,x_2,...,x_m are the zeros of x^m-1 and y_1,y_2,...,y_m the zeros of y^m+1 and m=2n-1.
In 1881, R. F. Scott posed a conjecture that the absolute value of permanent of square matrix with elements a(i,j)= (x_i - y_j)^(-1), where x_1,...,x_n are roots of x^n=1, while y_1,...,y_n are roots of y^n=-1, equals a((n-1)/2)/2^n, if n>=1 is odd, and 0, if n>=2 is even. After a century (in 1979), the conjecture was proved by H. Minc. - Vladimir Shevelev, Dec 01 2013
a(n) is the number of permutations in S_{2n+1} in which all cycles have odd length. - José H. Nieto S., Jan 09 2012
Number of 3-bundled increasing bilabeled trees with 2n labels. - Markus Kuba, Nov 18 2014
a(n) is the number of rooted, binary, leaf-labeled topologies with 2n+2 leaves that have n+1 cherry nodes. - Noah A Rosenberg, Feb 12 2019

			G.f. = 1 + 3*x + 45*x^2 + 1575*x^3 + 99225*x^4 + 9823275*x^5 + ...
M(5) =
  [1, 2, 3, 1, 5]
  [1, 2, 2, 4, 5]
  [1, 3, 3, 4, 5]
  [4, 2, 3, 4, 5]
  [1, 2, 3, 4, 5].
Integral_{x=0..oo} x^3*BesselK(1, sqrt(x)) = 1575*Pi. - _Olivier Gérard_, May 20 2009

Miklós Bóna, A walk through combinatorics, World Scientific, 2006.

Alois P. Heinz, Table of n, a(n) for n = 0..224
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
Guo-Niu Han and Christian Krattenthaler, Rectangular Scott-type permanents, arXiv:math/0003072 [math.RA], 2000.
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 17 Nov 2014.
MathOverflow, Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?, answer by Tom Copeland posted in Aug 2021.
Henryk Minc, On a conjecture of R. F. Scott (1881), Linear Algebra Appl., Vol. 28 (1979), pp. 141-153.
Theodoros Theodoulidis, On the Closed-Form Expression of Carson’s Integral, Period. Polytech. Elec. Eng. Comp. Sci., Vol. 59, No. 1 (2015), pp. 26-29.
Eric Weisstein's World of Mathematics, Struve function.

Cf. A001818, A000165.
Bisection of A000246, A053195, |A013069|, |A046126|. Cf. A000909.
Cf. A001044, A010791, |A129464|, A114779, are also values of similar moments.
Equals the row sums of A162005.
Cf. A316728.
Diagonal elements of A306364 in even-numbered rows.

I:=[1, 3]; [n le 2 select I[n] else (4*n^2-8*n+3)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Nov 18 2014

a:= n-> (d-> d(2*n-1)*d(2*n+1))(doublefactorial):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 30 2013
# second Maple program:
A079484 := n-> LinearAlgebra[Determinant](Matrix(2*n+1, (i, j)-> `if`(i+j=2*n+1, j, i))): seq(A079484(n), n=0..14); # Rainer Rosenthal, Jun 18 2024

a[n_] := (2n - 1)!!*(2n + 1)!!; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jan 30 2013 *)

/* Formula using the zeta function and a log integral:*/
L(n)= intnum(t=0, 1, log(1-1/t)^n);
Zetai(n)= -I*I^n*(2*Pi)^(n-1)/(n-1)*L(1-n);
a(m)={my(n=m+1);round(real(-I*2^(2*n-1)*Zetai(1/2-n)*L(-1/2+n)/(Zetai(-1/2+n)*L(1/2-n))))};
/* Gerry Martens, Mar 07 2011, adapted to offset 0 by Hugo Pfoertner, Jun 19 2024 */

{a(n) = if( n<0, -1 / self()(-1-n), (2*n + 1)! * (2*n)! / (n! * 2^n)^2 )}; /* Michael Somos, May 04 2017 */

{a(n) = if( n<0, -1 / self()(-1-n), my(m = 2*n + 1); m! * polcoeff( x / sqrt( 1 - x^2 + x * O(x^m) ), m))}; /* Michael Somos, May 04 2017 */

\\ using the Pochhammer symbol
a(n) = {my(P(x,k)=gamma(x+k)/gamma(x)); 4^n*round(P(1/2,n)*P(3/2,n))} \\ Hugo Pfoertner, Jun 20 2024

\\ Scott's (1881) method
a(n) = {my(m=2*n+1, X = polroots(x^m-1), Y = polroots(x^m+1), M = matrix(m, m, i, j, 1/(X[i]-Y[j]))); (-1)^n * round(2^m * real(matpermanent(M)))}; \\ Hugo Pfoertner, Jun 23 2024

D-finite with recurrence a(n) = (4*n^2 - 1) * a(n-1) for all n in Z.
a(n) = A001147(n)*A001147(n+1).
E.g.f.: 1/(1-x^2)^(3/2) (with interpolated zeros). - Paul Barry, May 26 2003
a(n) = (2n+1)! * C(2n, n) / 2^(2n). - Ralf Stephan, Mar 22 2004.
Alternatingly signed values have e.g.f. sqrt(1+x^2).
a(n) is the value of the n-th moment of (1/Pi)*BesselK(1, sqrt(x)) on the positive part of the real line. - Olivier Gérard, May 20 2009
a(n) = -2^(2*n-1)*exp(i*n*Pi)*gamma(1/2+n)/gamma(3/2-n). - Gerry Martens, Mar 07 2011
E.g.f. (odd powers) tan(arcsin(x)) = Sum_{n>=0} (2n-1)!!*(2n+1)!!*x^(2*n+1)/(2*n+1)!. - Vladimir Kruchinin, Apr 22 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - ((2*k+2)^2-1)/(1-x/(x - 1/G(k+1))); ( continued fraction ). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = (2^(2*n+3)*Gamma(n+3/2)*Gamma(n+5/2))/Pi. - Jean-François Alcover, Jul 20 2015
Limit_{n->oo} 4^n*(n!)^2/a(n) = Pi/2. - Daniel Suteu, Feb 05 2017
From Michael Somos, May 04 2017: (Start)
a(n) = (2*n + 1) * A001818(n).
E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = x / sqrt(1 - x^2) = tan(arcsin(x)).
Given e.g.f. A(x) = y, then x * y' = y + y^3.
a(n) = -1 / a(-1-n) for all n in Z.
0 = +a(n)*(+288*a(n+2) -60*a(n+3) +a(n+4)) +a(n+1)*(-36*a(n+2) -4*a(n+3)) +a(n+2)*(+3*a(n+2)) for all n in Z. (End)
a(n) = Sum_{k=0..2n} (k+1) * A316728(n,k). - Alois P. Heinz, Jul 12 2018
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_1(1)*Pi/2, where L is the modified Struve function.
Sum_{n>=0} (-1)^n/a(n) = 1 - H_1(1)*Pi/2, where H is the Struve function. (End)

Simpler description from Daniel Flath (deflath(AT)yahoo.com), Mar 05 2004

A084939 Pentagorials: n-th polygorial for k=5.

Original entry on oeis.org

1, 1, 5, 60, 1320, 46200, 2356200, 164934000, 15173928000, 1775349576000, 257425688520000, 45306921179520000, 9514453447699200000, 2350070001581702400000, 674470090453948588800000, 222575129849803034304000000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Robert Israel, Table of n, a(n) for n = 0..243
M. A. Asiru, Sequence factorial of g-gonal numbers, Int. J. Math. Educ. Sci. Technol., 44(4) (2012), 579-586.
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A000326, A006472, A001044, A000680, A084940, A084941, A084942, A084943, A084944, A085356.

a := n->(n!/2^n)*mul(3*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Table[k! Pochhammer[2/3, k] (3/2)^k, {k, 0, 20}] (* Jan Mangaldan, Mar 20 2013 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[5, #] &, 17, 0] (* Robert G. Wilson v, Dec 17 2016 *)

a(n)=n!/2^n*prod(i=1,n,3*i-1) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = polygorial(n, 5) = (A000142(n)/A000079(n))*A008544(n) = (n!/2^n)*Product_{i=0..n-1} (3*i+2) = (n!/2^n)*3^n*Pochhammer(2/3, n) = (n!/2^n)*3^n*GAMMA(n+2/3)/GAMMA(2/3).
a(n) ~ Gamma(1/3) * 3^(n+1/2) * n^(2*n+2/3) / (2^n * exp(2*n)). - Vaclav Kotesovec, Jul 17 2015
D-finite with recurrence a(n+1) = ((n+1)*(3*n+2)/2)*a(n) = A000326(n+1)*a(n). - Muniru A Asiru, Apr 05 2016
E.g.f.: hypergeom([2/3, 1], [], (3/2)*x). - Robert Israel, Apr 05 2016

A144084 T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 72, 96, 24, 1, 25, 200, 600, 600, 120, 1, 36, 450, 2400, 5400, 4320, 720, 1, 49, 882, 7350, 29400, 52920, 35280, 5040, 1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320

Offset: 0

Abdullahi Umar, Sep 10 2008, Sep 30 2008

T(n,k) is also the number of elements in the Green's J equivalence classes in the symmetric inverse monoid, I sub n.
T(n,k) is also the number of ways to place k nonattacking rooks on an n X n chessboard. It can be obtained by performing P(n,k) permutations of n-columns over each C(n,k) combination of n-rows for the given k-rooks. The rule is also applicable for unequal (m X n) sized rectangular boards. - Antal Pinter, Nov 12 2014
Rows also give the coefficients of the matching-generating polynomial of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Rows also give the coefficients of the independence polynomial of the n X n rook graph and clique polynomial of the n X n rook complement graph. - Eric W. Weisstein, Jun 13 and Sep 14 2017
T(n,k) is the number of increasing subsequences of length n-k over all permutations of [n]. - Geoffrey Critzer, Jan 08 2023

			T(3,1) = 9 because there are exactly 9 partial bijections (on a 3-element set) of height 1, namely: (1)->(1), (1)->(2), (1)->(3), (2)->(1), (2)->(2), (2)->(3), (3)->(1), (3)->(2), (3)->(3).
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   2;
  1,  9,  18,    6;
  1, 16,  72,   96,   24;
  1, 25, 200,  600,  600,  120;
  1, 36, 450, 2400, 5400, 4320, 720;
  ...

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, page 61.
J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995).
Vaclav Kotesovec, Non-attacking chess pieces, 6th ed. (2013), p. 216, p. 218.

Alois P. Heinz, Rows n = 0..140, flattened
Wayne A. Johnson, Exponential Hilbert series of equivariant embeddings, arXiv:1804.04943 [math.RT], 2018.
W. D. Munn, The characters of the symmetric inverse semigroup, Proc. Cambridge Philos. Soc. 53 (1957), 13-18.
Eric Weisstein's World of Mathematics, Clique Polynomial
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Rook Graph

T(n,k) = |A021010|. Sum of rows of T(n,k) is A002720. T(n,n) is the order of the symmetric group on an n-element set, n!.
Columns k=2..10 are A163102, A179058, A179059, A179060, A179061, A179062, A179063, A179064, A179065.
Cf. A089231, A105219, A295383.

/* As triangle */ [[(Binomial(n,k)^2)*Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jun 13 2017

T:= (n, k)-> (binomial(n, k)^2)*k!:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 04 2012

Table[Table[Binomial[n, k]^2 k!,{k, 0, n}], {n, 0, 6}] // Flatten (* Geoffrey Critzer, Dec 04 2012 *)
Table[ CoefficientList[n!*LaguerreL[n, x], x] // Abs // Reverse, {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 18 2013 *)
CoefficientList[Table[n! x^n LaguerreL[n, -1/x], {n, 0, 8}], x] // Flatten (* Eric W. Weisstein, Apr 24 2017 *)
CoefficientList[Table[(-x)^n HypergeometricU[-n, 1, -(1/x)], {n, 5}],
  x] // Flatten (* Eric W. Weisstein, Jun 13 2017 *)

T(n,k) = k! * binomial(n,k)^2 \\ Andrew Howroyd, Feb 13 2018

T(n,k) = (C(n,k)^2)*k!.
T(n,k) = A007318(n,k) * A008279(n,k). - Antal Pinter, Nov 12 2014
From Peter Bala, Jul 04 2016: (Start)
G.f.: exp(x*t)*I_0(2*sqrt(x)) = 1 + (1 + t)*x/1!^2 + (1 + 4*t + 2*t^2)*x^2/2!^2 + (1 + 9*t + 18*t^2 + 6*t^3)*x^3/3!^2 + ..., where I_0(x) = Sum_{n >= 0} (x/2)^(2*n)/n!^2 is a modified Bessel function of the first kind.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*n!/k!*binomial(n,k) *t^(n-k)*R(k,t). Cf. A089231. (End)
From Peter Bala, Oct 05 2019: (Start)
E.g.f.: 1/(1 - t*x)*exp(x/(1 - t*x)).
Recurrence for row polynomials: R(n+1,t) = (1 + (2*n+1)*t)R(n,t) - n^2*t^2*R(n-1,t), with R(0,t) = 1 and R(1,t) = 1 + t.
R(n,t) equals the denominator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The numerator polynomial is the (n+1)-th row polynomial of A089231. (End)
Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/A001044(n) = exp(y*x)*E(x) where E(x) = Sum_{n>=0} x^n/A001044(n). - Geoffrey Critzer, Jan 08 2023
Sum_{k=0..n} k*T(n,k) = A105219(n). - Alois P. Heinz, Jan 08 2023
T(n,k) = Sum_{d=0..2*k} c(k,d)*n^d, where c(k,d) = Sum_{j=max(d-k,0)..k} binomial(k,j)*A008275(k+j,d)/j!. - Eder G. Santos, Jan 23 2025

cp's OEIS Frontend

A007407 a(n) = denominator of Sum_{k=1..n} 1/k^2.

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A008955 Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

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A088311 Number of sets of lists with distinct list sizes, cf. A000262.

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A001819 Central factorial numbers: second right-hand column of triangle A008955.

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A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.

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