A074962 -id:A074962 - cp's OEIS frontend

A001142 a(n) = Product_{k=1..n} k^(2k - 1 - n).

1, 1, 2, 9, 96, 2500, 162000, 26471025, 11014635520, 11759522374656, 32406091200000000, 231627686043080250000, 4311500661703860387840000, 209706417310526095716965894400, 26729809777664965932590782608648192

Offset: 0

N. J. A. Sloane

Absolute value of determinant of triangular matrix containing binomial coefficients.
These are also the products of consecutive horizontal rows of the Pascal triangle. - Jeremy Hehn (ROBO_HEN5000(AT)rose.net), Mar 29 2007
Limit_{n->oo} a(n)*a(n+2)/a(n+1)^2 = e, as follows from lim_{n->oo} (1 + 1/n)^n = e. - Harlan J. Brothers, Nov 26 2009
A000225 gives the positions of odd terms. - Antti Karttunen, Nov 02 2014
Product of all unreduced fractions h/k with 1 <= k <= h <= n. - Jonathan Sondow, Aug 06 2015
a(n) is a product of the binomial coefficients from the n-th row of the Pascal triangle, for n= 0, 1, 2, ... For n > 0, a(n) means the number of all maximum chains in the poset formed by the n-dimensional Boolean cube {0,1}^n and the relation "precedes by weight". This relation is defined over {0,1}^n as follows: for arbitrary vectors u, v of {0,1}^n we say that "u precedes by weight v" if wt(u) < wt(v) or if u = v, where wt(u) denotes the (Hamming) weight of u. For more details, see the sequence A051459. - Valentin Bakoev, May 17 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
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).

T. D. Noe, Table of n, a(n) for n = 0..50
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
Harlan J. Brothers, Finding e in Pascal's Triangle, Mathematics Magazine, 85 (2012), p. 51.
Harlan J. Brothers, Pascal's Triangle: The Hidden Stor-e, The Mathematical Gazette, 96 (2012), 145-148.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.
Leroy Quet, Problem 1636, Mathematics Magazine, Dec. 2001, p. 403.

Cf. A000178, A002109, A007318, A000225, A056077, A249421, A187059 (2-adic valuation), A249343, A249345, A249346, A249347, A249151.

Cf. also A004788, A056606 (squarefree kernel), A256113.

List([0..15], n-> Product([0..n], k-> Binomial(n,k) )); # G. C. Greubel, May 23 2019

a001142 = product . a007318_row -- Reinhard Zumkeller, Mar 16 2015

[(&*[Binomial(n,k): k in [0..n]]): n in [0..15]]; // G. C. Greubel, May 23 2019

a:=n->mul(binomial(n,k), k=0..n): seq(a(n), n=0..14); # Zerinvary Lajos, Jan 22 2008

Table[Product[k^(2*k - 1 - n), {k, n}], {n, 0, 20}] (* Harlan J. Brothers, Nov 26 2009 *)
Table[Hyperfactorial[n]/BarnesG[n+2], {n, 0, 20}] (* Peter Luschny, Nov 29 2015 *)
Table[Product[(n - k + 1)^(n - 2 k + 1), {k, 1, n}], {n, 0, 20}] (* Harlan J. Brothers, Aug 26 2023 *)

a(n):= prod(binomial(n,k),k,0,n); n : 15; for i from 0 thru n do print (a(i)); /* Valentin Bakoev, May 17 2019 */

for(n=0,16,print(prod(m=1,n,binomial(n,m))))

A001142(n) = prod(k=1, n, k^((k+k)-1-n)); \\ Antti Karttunen, Nov 02 2014

from math import factorial, prod
from fractions import Fraction
def A001142(n): return prod(Fraction((k+1)**k,factorial(k)) for k in range(1,n)) # Chai Wah Wu, Jul 15 2022

a = lambda n: prod(k^k/factorial(k) for k in (1..n))
[a(n) for n in range(20)] # Peter Luschny, Nov 29 2015

(define (A001142 n) (mul (lambda (k) (expt k (+ k k -1 (- n)))) 1 n))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
;; Antti Karttunen, Oct 28 2014

a(n) = C(n, 0)*C(n, 1)* ... *C(n, n).
From Harlan J. Brothers, Nov 26 2009: (Start)
a(n) = Product_{j=1..n-2} Product_{k=1..j} (1+1/k)^k, n >= 3.
a(1) = a(2) = 1, a(n) = a(n-1) * Product_{k=1..n-2} (1+1/k)^k. (End)
a(n) = hyperfactorial(n)/superfactorial(n) =  A002109(n)/A000178(n). - Peter Luschny, Jun 24 2012
a(n) ~ A^2 * exp(n^2/2 + n - 1/12) / (n^(n/2 + 1/3) * (2*Pi)^((n+1)/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015
a(n) = Product_{i=1..n} Product_{j=1..i} (i/j). - Pedro Caceres, Apr 06 2019
a(n) = Product_{k=1..n} (n - k + 1)^(n - 2*k + 1). - Harlan J. Brothers, Aug 26 2023

More terms from James Sellers, May 01 2000

Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

Original entry on oeis.org

1, 2, 6, 16, 38, 88, 196, 420, 878, 1794, 3584, 7032, 13572, 25792, 48352, 89512, 163774, 296444, 531234, 943072, 1659560, 2896376, 5015700, 8622108, 14718652, 24960138, 42062200, 70458160, 117349856, 194381704, 320295312, 525123604

Offset: 0

R. J. Mathar, Feb 11 2009

Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
The inverse Euler transform is apparently 2, 3, 6, 6, 10, 9, 14, 12, 18, 15, 22, 18, 26, 21, ..., A016825 interlaced with A008585. - R. J. Mathar, Apr 23 2009
In general, for m >= 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m*k), then a(n) ~ exp(m/12 + 3/2 * (7*m*Zeta(3)/2)^(1/3) * n^(2/3)) * m^(1/6 + m/36) * (7*Zeta(3))^(1/6 + m/36) / (A^m * 2^(2/3 + m/9) * sqrt(3*Pi) * n^(2/3 + m/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Ali H. Al-Saedi, Congruences for restricted plane overpartitions modulo 4 and 8, Raman. J. 48 (2) (2019) 251
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.
Mirjana Vuletic, A generalization of MacMahon's formula, Trans. Am. Math. Soc. 361 (2009) 2789-2804.

Cf. A000219, A015128, A026007, A261384, A261386, A261389.
Cf. A206622, A206623, A206624, A261519, A261520.
Cf. A285462, A285447, A285460, A285461.
Cf. A285446, A285458, A285459.
Cf. A306081.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m,2)-sigma(m,2))/2*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, May 01 2010

Convolve A000219 with A026007.
O.g.f.: exp( Sum_{n>=1} (sigma_2(2n) - sigma_2(n))/2 *x^n/n ), where sigma_2(n) is the sum of squares of divisors of n (A001157). - Paul D. Hanna, May 01 2010
a(n) ~ exp(1/12 + 3 * 2^(-4/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122 and A302237. - Peter Bala, Dec 23 2021

A073009 Decimal expansion of Sum_{n >= 1} 1/n^n.

Original entry on oeis.org

1, 2, 9, 1, 2, 8, 5, 9, 9, 7, 0, 6, 2, 6, 6, 3, 5, 4, 0, 4, 0, 7, 2, 8, 2, 5, 9, 0, 5, 9, 5, 6, 0, 0, 5, 4, 1, 4, 9, 8, 6, 1, 9, 3, 6, 8, 2, 7, 4, 5, 2, 2, 3, 1, 7, 3, 1, 0, 0, 0, 2, 4, 4, 5, 1, 3, 6, 9, 4, 4, 5, 3, 8, 7, 6, 5, 2, 3, 4, 4, 5, 5, 5, 5, 8, 8, 1, 7, 0, 4, 1, 1, 2, 9, 4, 2, 9, 7, 0, 8, 9, 8, 4, 9, 9

Offset: 1

Robert G. Wilson v, Aug 03 2002

			1.291285997062663540407282590595600541498619368...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

Kenny Lau, Table of n, a(n) for n = 1..10001
Johan Bernoulli, Demonstratio Methodi Analyticae, qua usus est pro determinanda aliqua Quadratura exponentiali per seriem, Actis Eruditorum A (1697), p. 131. Collected in Opera Omnia, vol. 3, 1742. See p. 376ff.
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Jaroslav Hančl and Simon Kristensen, Metrical irrationality results related to values of the Riemann zeta-function, arXiv:1802.03946 [math.NT], 2018.
Randall Munroe, Approximations, xkcd Web Comic #1047, Apr 25 2012.
Simon Plouffe, Sum(1/n^n, n=1..infinity). [internet archive]
Eric Weisstein's World of Mathematics, Power Tower.
Eric Weisstein's World of Mathematics, Sophomore's Dream.

Cf. A077178 (continued fraction expansion).

Cf. A083648, A229191, A245637.

evalf(Sum(1/n^n, n=1..infinity), 120); # Vaclav Kotesovec, Jun 24 2016

RealDigits[N[Sum[1/n^n, {n, 1, Infinity}], 110]] [[1]]

suminf(n=1,n^-n) \\ Charles R Greathouse IV, Apr 25 2012

Equals Integral_{x = 0..1} dx/x^x.
Constant also equals the double integral Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y)^(x*y) dx dy. - Peter Bala, Mar 04 2012
Approximately log(3)^e, see Munroe link. - Charles R Greathouse IV, Apr 25 2012
Another approximation is A + A^(-19), where A is Glaisher-Kinkelin constant (A074962). - Noam Shalev, Jan 16 2015
From Petros Hadjicostas, Jun 29 2020: (Start)
Equals -Integral_{x=0..1, y=0..1} dx dy/((x*y)^(x*y)*log(x*y)). (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the integral Integral_{x = 0..1} dx/x^x.)
Equals -Integral_{x=0..1} log(x)/x^x dx. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the double integral of Peter Bala above.) (End)

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

Original entry on oeis.org

1, 1, 4, 10, 26, 59, 141, 310, 692, 1483, 3162, 6583, 13602, 27613, 55579, 110445, 217554, 424148, 820294, 1572647, 2992892, 5652954, 10605608, 19765082, 36609945, 67405569, 123412204, 224728451, 407119735, 733878402, 1316631730, 2351322765, 4180714647, 7401898452, 13051476707, 22922301583, 40105025130, 69909106888, 121427077241, 210179991927, 362583131144

Offset: 0

N. J. A. Sloane

Number of partitions of n if there are k(k+1)/2 kinds of k (k=1,2,...). E.g., a(3)=10 because we have six kinds of 3, three kinds of 2+1 because there are three kinds of 2 and 1+1+1+1. - Emeric Deutsch, Mar 23 2005
Euler transform of the triangular numbers 1,3,6,10,...
Equals A028377: [1, 1, 3, 9, 19, 46, 100, ...] convolved with the aerated version of A000294: [1, 0, 1, 0, 4, 0, 10, 0, 26, 0, 59, ...]. - Gary W. Adamson, Jun 13 2009
The formula for p3(n) in the article by S. Finch (page 2) is incomplete, terms with n^(1/2) and n^(1/4) are also needed. These terms are in the article by Mustonen and Rajesh (page 2) and agree with my results, but in both articles the multiplicative constant is marked only as C, resp. c3(m). The following is a closed form of this constant: Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8)) / (A^(1/2) * 2^(157/96) * 15^(13/96)) = A255939 = 0.213595160470..., where A = A074962 is the Glaisher-Kinkelin constant and Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015 [The new version of "Integer Partitions" by S. Finch contains the missing terms, see pages 2 and 5. - Vaclav Kotesovec, May 12 2015]
Number of solid partitions of corner-hook volume n (see arXiv:2009.00592 among links for definition). E.g., a(2) = 1 because there is only one solid partition [[[2]]] with cohook volume 2; a(3) = 4 because we have three solid partitions with two 1's (entry at (1,1,1) contributes 1, another entry at (2,1,1) or (1,2,1) or (1,1,2) contributes 2 to corner-hook volume) and one solid partition with single entry 3 (which contributes 3 to the corner-hook volume). Namely as 3D arrays [[[1],[1]]],[[[1]],[[1]]],[[[1]],[[1]]], [[[3]]]. - Alimzhan Amanov, Jul 13 2021

R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139. [Annotated scanned copy]
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, J. Stat. Phys. 158 (2015) 950-967; arXiv:1406.5605 [cond-mat.stat-mech], 2014.
Steven Finch, Integer Partitions, September 22, 2004, page 2. [Cached copy, with permission of the author]
Vaclav Kotesovec, Graph - The asymptotic ratio
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, J. Phys. A 36 (2003), no. 24, 6651-6659; arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314. [Annotated scanned copy]
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000293, A007294, A007326, A255939, A028377.
Cf. also A000041, A000219, A000335, A000391, A000417, A000428, A255965.
Cf. also A278403 (log of o.g.f.).

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> n*(n+1)/2): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 08 2008

a[0] = 1; a[n_] := a[n] = 1/(2*n)*Sum[(DivisorSigma[2, k]+DivisorSigma[3, k])*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)/2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 11 2015 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^3/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) = (1/(2*n))*Sum_{k=1..n} (sigma[2](k)+sigma[3](k))*a(n-k). - Vladeta Jovovic, Sep 17 2002
a(n) ~ Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4)*Pi^5) + 15^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2)*Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3*15^(1/4))) / (A^(1/2) * 2^(157/96) * 15^(13/96) * n^(61/96)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 11 2015
G.f.: exp(Sum_{k>=1} (sigma_2(k) + sigma_3(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 21 2018

More terms from Sascha Kurz, Aug 15 2002

A061256 Euler transform of sigma(n), cf. A000203.

Original entry on oeis.org

1, 1, 4, 8, 21, 39, 92, 170, 360, 667, 1316, 2393, 4541, 8100, 14824, 26071, 46422, 80314, 139978, 238641, 408201, 686799, 1156062, 1920992, 3189144, 5238848, 8589850, 13963467, 22641585, 36447544, 58507590, 93334008, 148449417, 234829969, 370345918

Offset: 0

Vladeta Jovovic, Apr 21 2001

This is also the number of ordered triples of permutations f, g, h in Symm(n) which all commute, divided by n!. This was conjectured by Franklin T. Adams-Watters, Jan 16 2006, and proved by J. R. Britnell in 2012.
According to a message on a blog page by "Allan" (see Secret Blogging Seminar link) it appears that a(n) = number of conjugacy classes of commutative ordered pairs in Symm(n).
John McKay (email to N. J. A. Sloane, Apr 23 2013) observes that A061256 and A006908 coincide for a surprising number of terms, and asks for an explanation. - N. J. A. Sloane, May 19 2013

			1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 92*x^6 + 170*x^7 + 360*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
J. R. Britnell, A formal identity involving commuting triples of permutations, arXiv:1203.5079 [math.CO], 2012.
J. R. Britnell, A formal identity involving commuting triples of permutations, Preprint 2012. - _N. J. A. Sloane_, Jun 13 2012
J. R. Britnell, A formal identity involving commuting triples of permutations, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013.
E. Marberg, How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system, arXiv preprint arXiv:1202.1311 [math.RT], 2012. - _N. J. A. Sloane_, Jun 10 2012
Secret Blogging Seminar, A peculiar numerical coincidence.
N. J. A. Sloane, Transforms
Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), this sequence (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A000203, A001001, A001970, A053529, A061255, A061257, A006908, A192065.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 18 2012 *)
nmax = 40; CoefficientList[Series[Product[1/QPochhammer[x^k]^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^j)^j); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^m+x*O(x^n))^2/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

a(n) = A072169(n) / n!.
G.f.: Product_{k=1..infinity} (1 - x^k)^(-sigma(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*sigma(d), cf. A001001.
G.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)^2 /n ). [Paul D. Hanna, Mar 28 2009]
G.f.: exp( Sum_{n>=1} sigma_2(n)*x^n/(1-x^n)/n ). [Vladeta Jovovic, Mar 28 2009]
G.f.: prod(n>=1, E(x^n)^n ) where E(x) = prod(k>=1, 1-x^k). [Joerg Arndt, Apr 12 2013]
a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2 - Pi^(4/3) * n^(1/3) / (4 * 3^(2/3) * Zeta(3)^(1/3)) - 1/24 - Pi^2/(288*Zeta(3))) * A^(1/2) * Zeta(3)^(11/72) / (2^(11/24) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 23 2018

Entry revised by N. J. A. Sloane, Jun 13 2012

A084448 Decimal expansion of (negative of) Kinkelin constant.

Original entry on oeis.org

1, 6, 5, 4, 2, 1, 1, 4, 3, 7, 0, 0, 4, 5, 0, 9, 2, 9, 2, 1, 3, 9, 1, 9, 6, 6, 0, 2, 4, 2, 7, 8, 0, 6, 4, 2, 7, 6, 4, 0, 3, 6, 3, 8, 0, 3, 3, 5, 2, 0, 1, 7, 8, 3, 6, 6, 6, 5, 2, 2, 3, 0, 6, 3, 5, 7, 3, 5, 9, 6, 9, 9, 6, 6, 6, 5, 7, 7, 1, 7, 2, 7, 5, 9, 5, 2, 5, 1, 0, 0, 3, 3, 2, 5, 0, 8, 7, 5, 5

Offset: 0

N. J. A. Sloane, Jun 27 2003

Named after the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 16 2021

			-0.1654211437004509292139196602427806427640363803352017836665223...

Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., Vol. 7, No. 4 (1998), pp. 343-359.
Hermann Kinkelin, Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math., Vol. 57 (1860), pp. 122-158; alternative link. See eq. (22), p. 133.
E. M. Wright, Asymptotic partition formulae, I: Plane partitions, Quart. J. Math., Vol. 2 (1931), pp. 177-189.

Cf. A001620, A074962, A084539.

Maple
```
Digits := 200; evalf(Zeta(1,-1));
```

RealDigits[1/12 - Log[Glaisher], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)

-zeta'(-1) \\ Charles R Greathouse IV, Dec 12 2013

Zeta(1, -1). Almkvist gives many formulas.
Equals (1 - gamma - log(2*Pi))/12 + Zeta'(2)/(2*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015
From Amiram Eldar, Jun 16 2021: (Start)
Equals 1/24 - gamma/3 - Sum_{k>=1} (zeta(2*k+1)-1)/((2*k+1)*(2*k+3)) = 1/12 - log(A), where A is the Glaisher-Kinkelin constant (A074962) (Kinkelin, 1860).
Equals 2 * Integral_{x>=0} x*log(x)/(exp(2*Pi*x)-1) dx = 2*A261819. (Wright, 1931). (End)

A005380 Expansion of 1 / Product_{k>=1} (1-x^k)^(k+1).

Original entry on oeis.org

1, 2, 6, 14, 33, 70, 149, 298, 591, 1132, 2139, 3948, 7199, 12894, 22836, 39894, 68982, 117948, 199852, 335426, 558429, 922112, 1511610, 2460208, 3977963, 6390942, 10206862, 16207444, 25596941, 40214896, 62868772, 97814358

Offset: 0

N. J. A. Sloane

Also, a(n) = number of partitions of the integer n where there are k+1 different kinds of part k for k = 1, 2, 3, ....
Also, a(n) = number of partitions of n objects of 2 colors. These are set partitions, the n objects are not labeled but colored, using two colors.  For each subset of size k there are k+1 different possibilities, i=0..k white and k-i black objects.
Also, a(n) = number of simple unlabeled graphs with n nodes of 2 colors whose components are complete graphs. - Geoffrey Critzer, Sep 27 2012

			We represent each summand, k, in a partition of n as k identical objects. Then we color each object. We have no regard for the order of the colored objects.
a(3) = 14 because we have:  www; wwb; wbb; bbb; ww + w; ww + b;  wb + w; wb + b; bb + w; bb + b; w + w + w; w + w + b; w + b + b; b + b + b, where the 2 colors are black b and white w. - _Geoffrey Critzer_, Sep 27 2012
a(3) = 14 because we have:  3; 3'; 3''; 3'''; 2 + 1; 2 + 1';  2' + 1; 2' + 1'; 2'' + 1; 2'' + 1'; 1 + 1 + 1; 1 + 1 + 1'; 1 + 1' + 1'; 1' + 1' + 1', where a part k of different sorts is given as k, k', k'', etc. - _Joerg Arndt_, Mar 09 2015
From _Alois P. Heinz_, Mar 09 2015: (Start)
The a(4) = 33 = 5 + 9 + 6 + 8 + 5 partitions of 4 objects of 2 colors are:
5 partitions for the integer partition of 4 = 1 + 1 + 1 + 1:
  01: {{b}, {b}, {b}, {b}}
  02: {{b}, {b}, {b}, {w}}
  03: {{b}, {b}, {w}, {w}}
  04: {{b}, {w}, {w}, {w}}
  05: {{w}, {w}, {w}, {w}}
9 partitions for the integer partition of 4 = 1 + 1 + 2:
  06: {{b}, {b}, {b,b}}
  07: {{b}, {w}, {b,b}}
  08: {{w}, {w}, {b,b}}
  09: {{b}, {b}, {w,b}}
  10: {{b}, {w}, {w,b}}
  11: {{w}, {w}, {w,b}}
  12: {{b}, {b}, {w,w}}
  13: {{b}, {w}, {w,w}}
  14: {{w}, {w}, {w,w}}
6 partitions for the integer partition of 4 = 2 + 2:
  15: {{b,b}, {b,b}}
  16: {{b,b}, {w,b}}
  17: {{b,b}, {w,w}}
  18: {{w,b}, {w,b}}
  19: {{w,b}, {w,w}}
  20: {{w,w}, {w,w}}
8 partitions for the integer partition of 4 = 1 + 3:
  21: {{b}, {b,b,b}}
  22: {{w}, {b,b,b}}
  23: {{b}, {w,b,b}}
  24: {{w}, {w,b,b}}
  25: {{b}, {w,w,b}}
  26: {{w}, {w,w,b}}
  27: {{b}, {w,w,w}}
  28: {{w}, {w,w,w}}
5 partitions for the integer partition of 4 = 4:
  29: {{b,b,b,b}}
  30: {{w,b,b,b}}
  31: {{w,w,b,b}}
  32: {{w,w,w,b}}
  33: {{w,w,w,w}}
Some see number partitions, others see set partitions, ...
(End)
It is obvious from the example of _Alois P. Heinz_ that a(n) enumerates multi-set partitions of a multi-set of n elements of two kinds. In the case that there is only one kind, this reduces to the usual case of numerical partitions. In the case that all the n elements are distinct, then this reduces to the case of set partitions. - _Michael Somos_, Mar 09 2015
There are a(3) = 14 plane partitions of 6 with trace 3; of 7 with trace 4; of 8 with trace 5; etc. See a formula above with the Stanley Exercise 7.99. - _Wolfdieter Lang_, Mar 09 2015
From _Daniel Forgues_, Mar 09 2015: (Start)
The a(3) = 14 = 4 + 6 + 4 partitions of 3 objects of 2 colors are:
4 partitions for the integer partition of 3 = 1 + 1 + 1:
  01: {{b}, {b}, {b}}
  02: {{b}, {b}, {w}}
  03: {{b}, {w}, {w}}
  04: {{w}, {w}, {w}}
6 partitions for the integer partition of 3 = 1 + 2:
  05: {{b}, {b,b}}
  06: {{w}, {b,b}}
  07: {{b}, {w,b}}
  08: {{w}, {w,b}}
  09: {{b}, {w,w}}
  10: {{w}, {w,w}}
4 partitions for the integer partition of 3 = 3:
  11: {{b,b,b}}
  12: {{w,b,b}}
  13: {{w,w,b}}
  14: {{w,w,w}}
The a(2) = 6 = 3 + 3 partitions of 2 objects of 2 colors are:
3 partitions for the integer partition of 2 = 1 + 1:
  01: {{b}, {b}}
  02: {{b}, {w}}
  03: {{w}, {w}}
3 partitions for the integer partition of 2 = 2:
  04: {{b,b}}
  05: {{w,b}}
  06: {{w,w}}
The a(1) = 2 partitions of 1 object of 2 colors are:
2 partitions for the integer partition of 1 = 1:
  01: {{b}}
  02: {{w}}
a(0) = 1: the empty partition, since empty sum is 0.
Triangle(sort of, since n_th row has p(n) = A000041 terms):
  1:  2
  2:  3, 3
  3:  4, 6, 4
  4:  5, 9, 6, 8, 5
  5:  6, ?, ?, ?, ?, ?, 6
  6:  7, ?, ?, ?, ?, ?, ?, ?, ?, ?, 7
Can we find a recurrence relation? (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 7.99, p. 484 and pp. 548-549.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Carlos A. A. Florentino, Plethystic exponential calculus and permutation polynomials, arXiv:2105.13049 [math.CO], 2021. Mentions this sequence.
Vaclav Kotesovec, Graph - The asymptotic ratio
P. A. MacMahon, Memoir on symmetric functions of the roots of systems of equations, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
N. J. A. Sloane, Transforms
R. P. Stanley, Theory and Applications of Plane Partitions: Part 2, Studies in Appl. Math., 1 (1971), 259-279.
R. P. Stanley, The conjugate trace and trace of a plane partition, J. Combin. Theory, vol. A14 53-65 1973, esp. p. 64.

Row sums of A054225.
Column k=2 of A075196.
Cf. A000219, A219555, A217093, A255050, A255052, A089353, A298988.

mul( (1-x^i)^(-i-1),i=1..80); series(%,x,80); seriestolist(%);
# second Maple program:
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> n+1): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

max = 31; f[x_] = Product[ 1/(1-x^k)^(k+1), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 08 2011, after g.f. *)
etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n==0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; a = etr[#+1&]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

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

EULER transform of b(n) = n+1.
a(n) ~ Zeta(3)^(13/36) * exp(1/12 - Pi^4/(432*Zeta(3)) + Pi^2 * n^(1/3) / (3*2^(4/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 2^(23/36) * 3^(1/2) * Pi * n^(31/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 07 2015
a(n) = A089353(n+m, m), n >= 1, for each m >= n. a(0) =1. See the Stanley reference, Exercise 7.99. - Wolfdieter Lang, Mar 09 2015
G.f.: exp(Sum_{k>=1} (sigma_1(k) + sigma_2(k))*x^k/k). - Ilya Gutkovskiy, Aug 11 2018

Edited by Christian G. Bower, Sep 07 2002
New name from Joerg Arndt, Mar 09 2015
Restored 1995 name. - N. J. A. Sloane, Mar 09 2015

A005249 Determinant of inverse Hilbert matrix.

Original entry on oeis.org

1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000

Offset: 0

N. J. A. Sloane

a(n) = 1/determinant of M(n)*(-1)^floor(n/2) where M(n) is the n X n matrix m(i,j)=1/(i-j+n).

For n>=2, a(n) = Product k=1...(n-1) (2k+1) * C(2k,k)^2. This is a special case of the Cauchy determinant formula. A similar formula exists also for A067689. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002

			The matrix begins:
  1    1/2  1/3  1/4  1/5  1/6  1/7  1/8  ...
  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  ...
  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10 ...
  1/4  1/5  1/6  1/7  1/8  1/9  1/10 1/11 ...
  1/5  1/6  1/7  1/8  1/9  1/10 1/11 1/12 ...
  1/6  1/7  1/8  1/9  1/10 1/11 1/12 1/13 ...

Philip J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 288.
Jerry Glynn and Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..25
Man-Duen Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.
Richard K. Guy, Letter to N. J. A. Sloane, Sep 1986.
John E. Lauer, Letter to N. J. A. Sloane, Dec 1980.
Sajad Salami, On special matrices related to Cauchy and Toeplitz matrices, Instítuto da Matemática e Estatística, Universidade Estadual do Rio de Janeiro (Brazil, 2019).
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A000178, A000515, A067689, A060739, A069945, A056040, A163085, A074962.

List([0..10],n->Product([1..n-1],k->(2*k+1)*Binomial(2*k,k)^2)); # Muniru A Asiru, Jul 07 2018

H=: % @: >: @: (+/~) @: i.
det=: -/ .* NB. Roger Hui, Oct 12 2005

with(linalg): A005249 := n-> 1/det(hilbert(n));

Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 0, n - 1} ]], {n, 1, 10} ]
Table[Denominator[Det[HilbertMatrix[n]]], {n, 0, 12}]//Quiet (* L. Edson Jeffery, Aug 05 2014 *)
Table[BarnesG[2 n + 1]/BarnesG[n + 1]^4, {n, 0, 10}] (* Jan Mangaldan, Sep 22 2021 *)

a(n)=n^n*prod(k=1,n-1,(n^2-k^2)^(n-k))/prod(k=0,n-1,k!^2)

PARI
```
a(n)=if(n<0,0,1/matdet(mathilbert(n)))
```

a(n)=if(n<0,0,prod(k=0,n-1,(2*k)!*(2*k+1)!/k!^4))

def A005249(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i) for i in (1..2*n-1))
[A005249(i) for i in (0..10)] # Peter Luschny, Sep 18 2012

a(n) = n^n*(Product_{k=1..n-1} (n^2 - k^2)^(n-k))/Product_{k=0..n-1} k!^2. - Benoit Cloitre, Jan 15 2003
The reciprocal of the determinant of an n X n matrix whose element at T(i, j) is 1/(i+j-1).
a(n+1) = a(n)*A000515(n) = a(n)*(2*n+1)*binomial(2n,n)^2. - Enrique Pérez Herrero, Mar 31 2010 [In other words, the partial products of sequence A000515. - N. J. A. Sloane, Jul 10 2015]
a(n) = n!*Product_{i=1..2n-1} binomial(i,floor(i/2)) = n!*|A069945(n)|. - Peter Luschny, Sep 18 2012
a(n) = Product_{i=1..2n-1} A056040(i) = A163085(2*n-1). - Peter Luschny, Sep 18 2012
a(n) ~ A^3 * 2^(2*n^2 - n - 1/12) * n^(1/4) / (exp(1/4) * Pi^n), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, May 01 2015
a(n) = A000178(2*n-1)/A000178(n-1)^4, for n >= 1. - Amiram Eldar, Oct 20 2022

1 more term from Jud McCranie, Jul 16 2000

Additional comments from Robert G. Wilson v, Feb 06 2002

A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).

Original entry on oeis.org

1, 1, 3, 60, 12600, 37837800, 2053230379200, 2431106898187968000, 73566121315513295589120000, 65191584694745586153436251091200000, 1906765806522767212441719098019963758016000000, 2048024348726152339387799085049745725891853852479488000000

Offset: 0

Clark Kimberling

Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing. Also number of ways to choose groups of 1, 2, 3, ..., n-1 and n objects out of n*(n+1)/2 objects. - Floor van Lamoen, Jul 16 2001
a(n) is the number of ways to linearly order the multiset {1,2,2,3,3,3,...n,n,...n}. - Geoffrey Critzer, Mar 08 2009
Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

			From _Gus Wiseman_, Aug 12 2020: (Start)
The a(3) = 60 permutations of the prime indices of A006939(3) = 360:
  (111223)  (121123)  (131122)  (212113)  (231211)
  (111232)  (121132)  (131212)  (212131)  (232111)
  (111322)  (121213)  (131221)  (212311)  (311122)
  (112123)  (121231)  (132112)  (213112)  (311212)
  (112132)  (121312)  (132121)  (213121)  (311221)
  (112213)  (121321)  (132211)  (213211)  (312112)
  (112231)  (122113)  (211123)  (221113)  (312121)
  (112312)  (122131)  (211132)  (221131)  (312211)
  (112321)  (122311)  (211213)  (221311)  (321112)
  (113122)  (123112)  (211231)  (223111)  (321121)
  (113212)  (123121)  (211312)  (231112)  (321211)
  (113221)  (123211)  (211321)  (231121)  (322111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..35
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Multinomial Coefficient

Cf. A000178, A014068, A022919, A052295, A208437, A327803.
A190945 counts the case of anti-run permutations.
A317829 counts partitions of this multiset.
A325617 is the version for factorials instead of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008480 counts permutations of prime indices.
A181818 gives products of superprimorials, with complement A336426.
Cf. A000142, A022559, A027423, A034841, A076954, A112798, A303279, A336417.

with(combinat):
a:= n-> multinomial(binomial(n+1, 2), $0..n):
seq(a(n), n=0..12);  # Alois P. Heinz, May 18 2013

Table[Apply[Multinomial ,Range[n]], {n, 0, 20}]  (* Geoffrey Critzer, Dec 09 2012 *)
Table[Multinomial @@ Range[n], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Length[Permutations[Join@@Table[i,{i,n},{i}]]],{n,0,4}] (* Gus Wiseman, Aug 12 2020 *)

a(n) = binomial(n+1,2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019

a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).
a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001.
a(n) = A052295(n)/A000178(n). - Lekraj Beedassy, Feb 19 2004
a(n) = A208437(n*(n+1)/2,n). - Alois P. Heinz, Apr 08 2016
a(n) ~ A * exp(n^2/4 + n + 1/6) * n^(n^2/2 + 7/12) / (2^((n+1)^2/2) * Pi^(n/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 02 2019
a(n) = A327803(n*(n+1)/2,n). - Alois P. Heinz, Sep 25 2019
a(n) = A008480(A006939(n)). - Gus Wiseman, Aug 12 2020

More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
More terms from Michel ten Voorde, Apr 12 2001
Better definition from L. Edson Jeffery, May 18 2013

A079478 Coefficient of x^0 in P(n,x) = (Product_{i=0..n-1} i!^2)/matdet(M(n)) of degree n^2 where M(n) is the n X n matrix m(i,j) = 1/(i+j+x).

Original entry on oeis.org

1, 2, 72, 172800, 60963840000, 5574884681318400000, 205619158526859285626880000000, 4394314874750658447092552646524928000000000, 73955304765761130113502867875624106401967636480000000000000

Offset: 0

Benoit Cloitre, Jan 15 2003

nonn

Product of all matrix elements of n X n matrix M(i,j) = i+j (i,j=1..n). - Alexander Adamchuk, Apr 12 2006

			Determinant of M(2) is 1/(x^4 + 12*x^3 + 53*x^2 + 102*x + 72) hence a(2)=72.

Alois P. Heinz, Table of n, a(n) for n = 0..20

Cf. A011379.

Central column in triangle A009963.

seq(mul(mul(k+j,j=1..n), k=1..n), n=0..8); # Zerinvary Lajos, Jun 01 2007

Table[Product[Product[(i+j),{i,1,n}],{j,1,n}],{n,0,10}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[BarnesG[2*n+2] / BarnesG[n+2]^2, {n,0,10}] (* Vaclav Kotesovec, Feb 28 2019 *)

a(n)=(n+1)*prod(i=0,n,(n+i)!)/prod(i=1,n+1,i!)

a(n) = prod(i=1, n, prod(j=1, n, i+j)); \\ Michel Marcus, Feb 27 2019

from math import prod, factorial
def A079478(n): return prod(i+j for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)<Chai Wah Wu, Nov 26 2023

a(n) = (n+1)*(Product_{i=0..n} (n+i)!)/Product_{i=1..n+1} i!.
a(n) = A000178(2n)/A000178(n)^2, i.e., "central supercombinations" by analogy with A000984. - Henry Bottomley, May 14 2005
a(n) = Product_{j=1..n} Product_{i=1..n} (i + j). - Alexander Adamchuk, Apr 12 2006
Asymptotic: a(n) ~ (2*n+1)^(2*n^2 + 2*n + 5/12)*(n+1)^(-n^2 - 2*n - 5/6) * exp(-zeta'(-1) - (3/2)*n^2 + 3/4)/(sqrt(2*Pi)). - Peter Luschny, Nov 26 2012
a(n) = BarnesG(2*n+2) / BarnesG(n+2)^2. - Vaclav Kotesovec, Feb 28 2019
a(n) ~ A * 2^(2*n*(n+1) - 1/12) * n^(n^2 - 5/12) / (sqrt(Pi) * exp(3*n^2/2 + 1/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 04 2023

cp's OEIS Frontend

A001142 a(n) = Product_{k=1..n} k^(2k - 1 - n).

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A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

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A073009 Decimal expansion of Sum_{n >= 1} 1/n^n.

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A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

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A061256 Euler transform of sigma(n), cf. A000203.

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Maple

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PARI

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A084448 Decimal expansion of (negative of) Kinkelin constant.

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A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).