[seqfan] Re: Breaking news on partition numbers.

[Georgi Guninski]

On Mon, Jan 24, 2011 at 06:22:57AM -0500, RGWv wrote:
> ?Neil,
> 
> They are fairly short.
> 
> Bob.
> 
> number of partitions of n having no element smaller than k.
> f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k
> == n, 1, f[n, k + 1] + f[n - k, k]]]]
> 
> number of partitions of n having no element greater than k.
> g[n_, 1] := 1; g[1, k_] := 1; g[n_, k_] := g[n, k] = If[k > n, g[n,
> k - 1], g[n, k - 1] + g[n - k, k]]
>

thank you.
i did some blackbox changes to your function and other sequences popped up.

changing:

|n-k| to |n-2*k| appears to give A000009 number of partitions of n into distinct parts; number of partitions of n into odd parts.
|k+1| to |2*k+1| appears to give A000929 Dimension of n-th degree part of Steenrod algebra
|k+1| to |2*k| appears to give A018819 Binary partition function: number of partitions of n into powers of 2
|k+1| to |next_prime(k)| appears to give A034891 Number of different products of partitions of n; partitions of n into prime parts (1 included) (verified to 500).