OFFSET
0,3
COMMENTS
Number of partitions of n with 1 kind of n's, 2 kinds of (n-1)'s, ..., n kinds of 1's, see example. [Joerg Arndt, May 17 2013]
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
FORMULA
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724158141653278798514832712869470973196907560641... and c = 0.2030089852709942695768237484498370155967795685257713505678384193773498... - Vaclav Kotesovec, Aug 21 2018
EXAMPLE
Let [x^n] F(x) denote the coefficient of x^n in F(x); then
a(0) = 1;
a(1) = [x] 1/(1-x) = 1;
a(2) = [x^2] 1/((1-x)^2*(1-x^2)) = 4;
a(3) = [x^3] 1/((1-x)^3*(1-x^2)^2*(1-x^3)) = 17;
a(4) = [x^4] 1/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) = 80; ...
as illustrated below.
The coefficients in Product_{k=1..n} 1/(1-x^k)^(n-k+1) for n=0..9 begin:
n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 2,(4), 6, 9, 12, 16, 20, 25, 30, 36, 42, ...]; (A002620)
n=3: [1, 3, 8, (17), 33, 58, 97, 153, 233, 342, 489, 681, ...]; (A002625)
n=4: [1, 4, 13, 34, (80), 170, 339, 636, 1141, 1964, 3270, ...];
n=5: [1, 5, 19, 58, 157,(384), 874, 1869, 3803, 7408, 13907, ...];
n=6: [1, 6, 26, 90, 273, 746, (1887), 4474, 10062, 21620, ...];
n=7: [1, 7, 34, 131, 438, 1314, 3632, (9385), 22940, 53466, ...];
n=8: [1, 8, 43, 182, 663, 2158, 6445, 17944, (47139), 117842, ...];
n=9: [1, 9, 53, 244, 960, 3361, 10757, 32008, 89651, (238488), ...]; ...
where the coefficients in parenthesis start this sequence.
Incidentally, the antidiagonal sums in the above table form A206119.
From Joerg Arndt, May 17 2013: (Start)
There are a(3)=17 partitions of 3 into 1 kind of 3's, 2 kinds of 2's, and 3 kinds of 1's:
01: [ 1:0 1:0 1:0 ]
02: [ 1:0 1:0 1:1 ]
03: [ 1:0 1:0 1:2 ]
04: [ 1:0 1:1 1:1 ]
05: [ 1:0 1:1 1:2 ]
06: [ 1:0 1:2 1:2 ]
07: [ 1:0 2:0 ]
08: [ 1:0 2:1 ]
09: [ 1:1 1:1 1:1 ]
10: [ 1:1 1:1 1:2 ]
11: [ 1:1 1:2 1:2 ]
12: [ 1:1 2:0 ]
13: [ 1:1 2:1 ]
14: [ 1:2 1:2 1:2 ]
15: [ 1:2 2:0 ]
16: [ 1:2 2:1 ]
17: [ 3:0 ]
(End)
MATHEMATICA
Table[SeriesCoefficient[Product[1/(1 - x^k)^(n-k+1), {k, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Aug 21 2018 *)
PROG
(PARI) {a(n)=polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^n))^(n-k+1)), n)}
for(n=0, 41, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2012
STATUS
approved