OFFSET
0,11
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
FORMULA
For n>0: a(n) = b(n, 1) where b(n, k) = if n>k*(k+1)/2 then b(n-k*(k+1)/2, k+1) + b(n, k+1) else (if n=k*(k+1)/2 then 1 else 0). - Reinhard Zumkeller, Aug 26 2003
a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1)*Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) * ((sqrt(2)-1)*Zeta(3/2))^(1/3) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Jan 02 2017
G.f.: prod_{i>=1} (1+x^A000217(i)). - R. J. Mathar, Sep 20 2020
EXAMPLE
a(31) counts these partitions: [28,3], [21,10], [21,6,3,1], [15,10,6] Clark Kimberling, Mar 09 2014
MATHEMATICA
Drop[ CoefficientList[ Series[ Product[(1 + x^(k*(k + 1)/2)), {k, 1, 15}], {x, 0, 102}], x], 1]
(* also *)
t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows unrestricted partitions*)
d[n_] := Select[p[n], Max[Length /@ Split@#] == 1 &]; Table[d[n], {n, 1, 31}] (*shows strict partitions*)
Table[Length[d[n]], {n, 1, 70}] (* Clark Kimberling, Mar 09 2014 *)
nmax = 100; nn = Floor[Sqrt[8*nmax + 1]/2] + 1; poly = ConstantArray[0, nn*(nn+1)/2 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k*(k+1)/2 + 1]], {j, nn*(nn+1)/2, k*(k+1)/2, -1}]; , {k, 2, nn}]; Take[poly, nmax + 1] (* Vaclav Kotesovec, Dec 10 2016 *)
PROG
(Haskell)
a024940 = p $ tail a000217_list where
p _ 0 = 1
p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Jun 28 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved