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A206226 Number of partitions of n^2 into parts not greater than n. 13
1, 1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574, 97299665768397, 800212617435074, 6617003142869419, 54985826573015541, 458962108485797208, 3846526994743330075 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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]
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
From Vaclav Kotesovec, May 25 2015: (Start)
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.
-------
j c(j)
1 0.1582087202672504149766310999238...
2 0.0794245035465730707705885572860...
3 0.0530017980244665552354063060738...
4 0.0397666338404544208556554596295...
5 0.0318193213988281353709268311928...
...
17 0.0093617308583114626385718275875...
c(j) for big j asymptotically approaches 1 / (2*Pi*j).
---------
j d(j)
1 9.15337019245412246194853029240... = A258268
2 16.57962120993269533568313969522...
3 23.98280768122086592445663786762...
4 31.37931997386325137074644287711...
5 38.77298550971449870728474612568...
...
17 127.45526806942537991146993713837...
d(j) for big j asymptotically approaches j * exp(2).
(End)
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
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)
FORMULA
a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).
a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - Vaclav Kotesovec, Sep 07 2014
MAPLE
T:= proc(n, k) option remember;
`if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
end:
seq(T(n^2, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz
MATHEMATICA
Table[SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}], {x, 0, n^2}], {n, 0, 20}] (* Vaclav Kotesovec, May 25 2015 *)
(* 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 *)
PROG
(PARI) {a(n)=polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^2)))), n^2)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
Column k=2 of A238016.
Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).
Sequence in context: A345883 A365122 A233397 * A371495 A326809 A326557
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2012
STATUS
approved

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Last modified July 14 05:06 EDT 2024. Contains 374291 sequences. (Running on oeis4.)