# [seqfan] Formula for A124577 ?

Paul D Hanna pauldhanna at juno.com
Sun Nov 15 00:54:48 CET 2009

```SeqFans,
A124577 (copied below) has only 5 terms.
These values coincide with the formula:
a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k) for n>0
(see A000000 below).

Can it be shown that these sequences are the same?

If not, I plan to submit A000000 as a new sequence and reference A124577.
Paul
-------------------------------------------------------------------------
A124577

1, 6, 39, 356, 4055

Define p(alpha) to be the number of H-conjugacy classes where H is a
Young subgroup of type alpha of the symmetric group S_n.
Then a(n) = sum p(alpha) where |alpha| = n and alpha has at most n parts.

-------------------------------------------------------------------------
A000000  -  NEW SEQUENCE (if it cannot be shown to equal A124577).

1,6,39,356,4055,57786,983535,19520264,441967518,11235798510,
316719689506,9800860032876,330230585628437,12032866998445818,
471416196117401340,19758835313514076176,882185444649249777913,

NAME.
a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k) for n>0.

FORMULA.
a(n) = Sum_{k=1..n} A008284(n,k)*n^k,
where A008284(n,k) = number of partitions of n
in which the greatest part is k, 1<=k<=n.

EXAMPLE:
a(n) = Sum_{k=1..n} A008284(n,k)*n^k :
a(1) = 1 ;
a(2) = 2 + 2^2 ;
a(3) = 3 + 3^2 + 3^3 ;
a(4) = 4 + 2*4^2 + 4^3 + 4^4 ;
a(5) = 5 + 2*5^2 + 2*5^3 + 5^4 + 5^5 ;
a(6) = 6 + 3*6^2 + 3*6^3 + 2*6^4 + 6^5 + 6^6 ;
a(7) = 7 + 3*7^2 + 4*7^3 + 3*7^4 + 2*7^5 + 7^6 + 7^7 ;
a(8) = 8 + 4*8^2 + 5*8^3 + 5*8^4 + 3*8^5 + 2*8^6 + 8^7 + 8^8 ; ...
-------------------------------------------------------------------------

```