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A024940
Number of partitions of n into distinct triangular numbers.
43
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 0, 2, 3, 1, 1, 3, 2, 1, 4, 3, 0, 3, 3, 2, 4, 3, 3, 3, 2, 3, 3, 2, 4, 6, 4, 2, 5, 4, 2, 6, 5, 3, 7, 6, 3, 5, 5, 5, 6, 5, 4, 7, 7, 6, 8, 6, 5, 9, 7, 4, 9, 9, 6, 10, 9, 4, 9, 10, 8, 11, 11, 9, 10, 10, 9, 10, 10, 9, 14, 14, 7, 14, 14, 7, 15, 15, 8, 15, 17, 13
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
STATUS
approved