cp's OEIS Frontend

A206226 Number of partitions of n^2 into parts not greater than n.

%I A206226 #50 Jun 11 2015 05:07:16
%S A206226 1,1,3,12,64,377,2432,16475,116263,845105,6292069,47759392,368379006,
%T A206226 2879998966,22777018771,181938716422,1465972415692,11902724768574,
%U A206226 97299665768397,800212617435074,6617003142869419,54985826573015541,458962108485797208,3846526994743330075
%N A206226 Number of partitions of n^2 into parts not greater than n.
%C A206226 Also the number of partitions of n^2 using n or fewer numbers. Thus for n=3 one has: 9; 1,8; 2,7; 3,6; 4,5; 1,1,7; 1,2,6; 1,3,5; 1,4,4; 2,2,5; 2,3,4; 3,3,3. - _J. M. Bergot_, Mar 26 2014 [computations done by _Charles R Greathouse IV_]
%C A206226 The partitions in the comments above are the conjugates of the partitions in the definition. By conjugation we have: "partitions into parts <= m" are equinumerous with "partitions into at most m parts". - _Joerg Arndt_, Mar 31 2014
%C A206226 From _Vaclav Kotesovec_, May 25 2015: (Start)
%C A206226 In general, "number of partitions of j*n^2 into parts that are at most n" is (for j>0) asymptotic to c(j) * d(j)^n / n^2, where c(j) and d(j) are a constants.
%C A206226 -------
%C A206226 j  c(j)
%C A206226 1  0.1582087202672504149766310999238...
%C A206226 2  0.0794245035465730707705885572860...
%C A206226 3  0.0530017980244665552354063060738...
%C A206226 4  0.0397666338404544208556554596295...
%C A206226 5  0.0318193213988281353709268311928...
%C A206226 ...
%C A206226 17 0.0093617308583114626385718275875...
%C A206226    c(j) for big j asymptotically approaches 1 / (2*Pi*j).
%C A206226 ---------
%C A206226 j    d(j)
%C A206226 1    9.15337019245412246194853029240... = A258268
%C A206226 2   16.57962120993269533568313969522...
%C A206226 3   23.98280768122086592445663786762...
%C A206226 4   31.37931997386325137074644287711...
%C A206226 5   38.77298550971449870728474612568...
%C A206226 ...
%C A206226 17 127.45526806942537991146993713837...
%C A206226    d(j) for big j asymptotically approaches j * exp(2).
%C A206226 (End)
%C A206226 d(j) = r^(2*j+1)/(r-1), where r is the root of the equation polylog(2, 1-r) + (j+1/2)*log(r)^2 = 0. - _Vaclav Kotesovec_, Jun 11 2015
%H A206226 Alois P. Heinz and Vaclav Kotesovec, <a href="/A206226/b206226.txt">Table of n, a(n) for n = 0..382</a> (first 150 terms from Alois P. Heinz)
%F A206226 a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).
%F A206226 a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - _Vaclav Kotesovec_, Sep 07 2014
%p A206226 T:= proc(n, k) option remember;
%p A206226       `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
%p A206226     end:
%p A206226 seq(T(n^2, n), n=0..20); # _Vaclav Kotesovec_, May 25 2015 after _Alois P. Heinz_
%t A206226 Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n^2}],{n,0,20}] (* _Vaclav Kotesovec_, May 25 2015 *)
%t A206226 (* A program to compute the constants d(j) *) Table[r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->60], {j, 1, 5}] (* _Vaclav Kotesovec_, Jun 11 2015 *)
%o A206226 (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2)))),n^2)}
%o A206226 for(n=0,25,print1(a(n),", "))
%Y A206226 Cf. A173519, A206227, A206240, A107379, A258268.
%Y A206226 Column k=2 of A238016.
%Y A206226 Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).
%K A206226 nonn
%O A206226 0,3
%A A206226 _Paul D. Hanna_, Feb 05 2012