A068980 - OEIS

%I #39 Aug 20 2024 01:16:37

%S 1,1,1,1,2,2,2,2,3,3,4,4,5,5,6,6,7,7,8,8,11,11,12,12,15,15,16,16,19,

%T 19,22,22,25,25,28,29,32,32,35,36,42,42,45,46,52,53,56,57,63,64,70,71,

%U 77,78,84,87,94,95,101,104,115,116,122,125,136,139,146,149,160,163,175

%N Number of partitions of n into nonzero tetrahedral numbers.

%H Vaclav Kotesovec, <a href="/A068980/b068980.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhicheng Gao, Andrew MacFie and Daniel Panario, <a href="https://doi.org/10.37236/630">Counting words by number of occurrences of some patterns</a>, The Electronic Journal of Combinatorics, 18 (2011), #P143.

%F G.f.: 1 / Product_{k>=3} (1 - z^binomial(k, 3)).

%F G.f.: Sum_{i>=0} x^(i*(i+1)*(i+2)/6) / Product_{j=1..i} (1 - x^(j*(j+1)*(j+2)/6)). - _Ilya Gutkovskiy_, Jun 08 2017

%e a(10) = 4 because we can write 10 = 10 = 4 + 4 + 1 + 1 = 4 + 1 + 1 + 1 + 1 + 1 = 1 + ... + 1.

%t nmax = 100; CoefficientList[Series[Product[1/(1-x^(k*(k+1)*(k+2)/6)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 09 2017 *)

%Y See also A007294 (partitions into triangular numbers), A000292 (tetrahedral numbers).

%Y Cf. A003108, A226748.

%K easy,nonn

%O 0,5

%A _Franklin T. Adams-Watters_, Apr 01 2002