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A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132. 3
1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 5, 5, 7, 7, 10, 10, 13, 13, 16, 18, 21, 25, 27, 32, 33, 41, 44, 53, 57, 65, 73, 81, 93, 102, 118, 128, 145, 159, 181, 200, 224, 246, 275, 304, 337, 375, 413, 460, 503, 559, 614, 679, 749, 821, 907, 991, 1096, 1197, 1319, 1442, 1582, 1733, 1893, 2076, 2265, 2482, 2702, 2956, 3220 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
FORMULA
G.f.: prod(n>=1, 1 + q^n ) / prod(n>=1, 1 + q^(n*(n+1)/2) ). [Joerg Arndt, Apr 01 2014]
a(n) ~ exp(Pi*sqrt(n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / (2+sqrt(2)) - 3*(3-2*sqrt(2)) * Zeta(3/2)^2 / (16*Pi)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jan 02 2017
EXAMPLE
a(10) = #{8+2} = 1;
a(11) = #{11, 9+2, 7+4, 5+4+2} = 4;
a(12) = #{12, 8+4, 7+5} = 3;
a(13) = #{13, 11+2, 9+4, 8+5, 7+4+2} = 5.
MAPLE
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, 1, add(b(n-i*j, i-1), j=0..min(n/i,
`if`(issqr(8*i+1), 0, 1)))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..80); # Alois P. Heinz, Apr 01 2014
MATHEMATICA
b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n==0, 1, Sum[b[n-i*j, i-1], {j, 0, Min[n/i, If[IntegerQ[Sqrt[8*i+1]], 0, 1]]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)
PROG
(Haskell)
a225045 = p a014132_list where
p _ 0 = 1
p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
(PARI) N=66; q='q+O('q^N); Vec( prod(n=1, N, 1 + q^n) / prod(n=1, N, 1 + q^(n*(n+1)/2)) ) \\ Joerg Arndt, Apr 01 2014
CROSSREFS
Sequence in context: A291323 A290988 A165025 * A361736 A278575 A333879
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 25 2013
STATUS
approved

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Last modified March 19 07:21 EDT 2024. Contains 370955 sequences. (Running on oeis4.)