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A002774 Number of bipartite partitions of n white objects and n black ones.
(Formerly M1925 N0760)
18
1, 2, 9, 31, 109, 339, 1043, 2998, 8406, 22652, 59521, 151958, 379693, 927622, 2224235, 5236586, 12130780, 27669593, 62229990, 138095696, 302673029, 655627975, 1404599867, 2977831389, 6251060785, 12999299705, 26791990052, 54750235190, 110977389012 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of ways to factor p^n*q^n where p and q are distinct primes.

REFERENCES

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, see p(n,n), page 778. - N. J. A. Sloane, Dec 30 2018

A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.

A. Murthy, Program for finding out the number of Smarandache factor partitions. (To be published in Smarandache Notions Journal).

Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.14.

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).

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..100 from Alois P. Heinz)

F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.

F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.

F. C. Auluck, On partitions of bipartite numbers, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 49, Issue 01, January 1953, pp. 72-83. (full article)

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956 (Annotated scanned pages from, plus a review)

FORMULA

a(n) = A054225(2n, n) = A091437(2n).

a(n) ~ Zeta(3)^(19/36) * exp(3*Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (6*Zeta(3)^(1/3)) + Zeta'(-1) - Pi^4/(432*Zeta(3))) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)). - Vaclav Kotesovec, Jan 30 2016

Formula (25) in the article by Auluck is incorrect. The correct formula is: p(n,n) ~ c^(19/12) * exp(3*c*n^(2/3) + 3*d*n^(1/3) + Zeta'(-1) - 3*d^2/(4*c)) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)), where c = Zeta(3)^(1/3), d = Zeta(2)/(3*c). Also formula (24) is incorrect. - Vaclav Kotesovec, Jan 30 2016

From Vaclav Kotesovec, Feb 04 2016: (Start)

The correct formula (24) is p(m,n) ~ c^(7/4)/(2*Pi*sqrt(3)) * exp(3*c*(m*n)^(1/3) + 3*d*(m+n)/(2*(m*n)^(1/3)) - 19*log(m*n)/24 - ((m/n - 2*n/m)*log(m) + (n/m - 2*m/n)*log(n))/36 - (m/n + n/m)*(log(c)/12 + Zeta'(-1) - 1/12 + 3*d^2/(4*c)) + 3*d^2/(4*c) - 3*log(2*Pi)/4 + fi((n/m)^(1/2))),

where m and n are of the same order, c = Zeta(3)^(1/3), d = Zeta(2)/(3*c) and fi(alfa) = Integral_{t=0..infinity} 1/t*(1/(exp(alfa*t)-1)/(exp(t/alfa)-1) - alfa/t/(exp(alfa*t)-1) - 1/alfa/t/(exp(t/alfa)-1) + 1/t^2 + 1/4/(exp(alfa*t)-1) + 1/4/(exp(t/alfa)-1) - alfa/4/t - 1/4/alfa/t).

If m = n then alfa = 1 and fi(1) = 3*Zeta'(-1) + log(2*Pi)/4 - 1/6.

For the asymptotic formula for fixed m see A054225.

(End)

MAPLE

with(numtheory):

b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0,

      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))

    end:

a:= n-> b(6^n$2):

seq(a(n), n=0..30);  # Alois P. Heinz, Jun 27 2013

MATHEMATICA

max = 26; se = Series[ Sum[ Log[1 - x^(n-k)*y^k], {n, 1, 2max}, {k, 0, n}], {x, 0, 2max}, {y, 0, 2max}]; coes = CoefficientList[ Series[ Exp[-se], {x, 0, 2max}, {y, 0, 2max}], {x, y}]; a[n_] := coes[[n+1, n+1]]; Table[a[n], {n, 0, max} ](* Jean-Fran├žois Alcover, Dec 06 2011 *)

CROSSREFS

Cf. A005380.

Cf. A219554. Column k=2 of A219727. - Alois P. Heinz, Nov 26 2012

Cf. also A000041, A000070, A000291, A000412, A000465, A000491, A002755, A002756, A002757, A002758, A002759, A277239.

Main diagonal of A054225 if that entry is drawn as a square array. - N. J. A. Sloane, Dec 30 2018

Sequence in context: A277244 A277245 A277246 * A318124 A295264 A150905

Adjacent sequences:  A002771 A002772 A002773 * A002775 A002776 A002777

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected using A000491.

Edited by Christian G. Bower, Jan 08 2004

STATUS

approved

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Last modified March 23 17:19 EDT 2019. Contains 321432 sequences. (Running on oeis4.)