A024940 Number of partitions of n into distinct triangular numbers.

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

Cf. A000217, A033461, A007294, A280366.

Sequence in context: A264997 A222759 A357072 * A324827 A205217 A054635

Adjacent sequences: A024937 A024938 A024939 * A024941 A024942 A024943

Keyword

nonn

Author

Clark Kimberling

Status

approved