A002759 - OEIS

%I M5278 N2295 #41 Jan 01 2024 08:01:07

%S 42,139,392,907,1941,3804,7128,12693,21893,36535,59521,94664,147794,

%T 226524,342006,508866,747753,1085635,1559725,2218272,3126541,4368724,

%U 6056705,8333955,11388614,15460291,20859497,27979454,37324367,49529018

%N Number of bipartite partitions of n white objects and 10 black ones.

%C Number of ways to factor p^n*q^10 where p and q are distinct primes.

%C a(n) is the number of multiset partitions of the multiset {r^n, s^10}. - _Joerg Arndt_, Jan 01 2024

%D 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. 2.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A002759/b002759.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..200 from Alois P. Heinz)

%H M. S. Cheema and H. Gupta, <a href="/A002755/a002755.pdf">Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956</a> (Annotated scanned pages from, plus a review)

%F a(n) = if n <= 10 then A054225(10,n) else A054225(n,10). - _Reinhard Zumkeller_, Nov 30 2011

%F a(n) ~ sqrt(3) * n^4 * exp(Pi*sqrt(2*n/3)) / (5600*Pi^10). - _Vaclav Kotesovec_, Feb 01 2016

%t p = 2; q = 3; b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[p^n*q^10, p^n*q^10]; Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)

%t nmax = 50; CoefficientList[Series[(42 + 55*x + 72*x^2 + 68*x^3 + 55*x^4 + 22*x^5 - 21*x^6 - 72*x^7 - 126*x^8 - 178*x^9 - 222*x^10 - 203*x^11 - 169*x^12 - 81*x^13 + 15*x^14 + 125*x^15 + 209*x^16 + 286*x^17 + 303*x^18 + 299*x^19 + 219*x^20 + 107*x^21 - 4*x^22 - 117*x^23 - 208*x^24 - 263*x^25 - 257*x^26 - 232*x^27 - 151*x^28 - 69*x^29 + 29*x^30 + 92*x^31 + 130*x^32 + 145*x^33 + 143*x^34 + 97*x^35 + 48*x^36 - 2*x^37 - 39*x^38 - 48*x^39 - 58*x^40 - 41*x^41 - 31*x^42 - 19*x^43 - 4*x^44 + 19*x^45 + 21*x^46 + 20*x^47 + 13*x^48 - 4*x^49 - 9*x^50 - 10*x^51 + 2*x^52 + 4*x^53 - x^54)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8) * (1-x^9) * (1-x^10)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 01 2016 *)

%Y Column 10 of A054225.

%Y Cf. A005380.

%K nonn

%O 0,1

%A _N. J. A. Sloane_

%E Edited by _Christian G. Bower_, Jan 08 2004