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A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3.

%I A025157 #47 Jan 26 2022 12:56:25
%S A025157 1,1,1,1,1,2,2,3,3,4,4,5,6,7,8,10,11,13,15,17,19,22,25,28,32,36,41,46,
%T A025157 52,58,66,73,82,91,102,113,126,139,155,171,190,209,232,255,282,310,
%U A025157 342,375,413,452,497,544,596,651,713,778,850,927,1011,1101,1200,1305,1420,1544,1677,1821,1977,2144,2324,2519,2728
%N A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3.
%C A025157 Also number of partitions of n into distinct parts in which the smallest part is greater than or equal to number of parts. - _Vladeta Jovovic_, Aug 06 2004
%H A025157 Alois P. Heinz, <a href="/A025157/b025157.txt">Table of n, a(n) for n = 0..10000</a>
%H A025157 I. Martinjak and D. Svrtan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Martinjak/mart3.html">New Identities for the Polarized Partitions and Partitions with d-Distant Parts</a>, J. Int. Seq. 17 (2014) # 14.11.4.
%F A025157 G.f.: sum(i>=1, x^(3*A000217(i)-2*i)/product(j=1..i, 1-x^j)). - _Jon Perry_, Jul 20 2004
%F A025157 G.f.: sum(n>=0, x^(n*(3*n-1)/2)/prod(k=1..n,1-x^k)). - _Joerg Arndt_, Jan 29 2011
%F A025157 a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*r*sqrt(Pi*(1+3*r^2))), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the root of the equation r^3 + r = 1 and c = 3*(log(r))^2/2 + polylog(2, 1-r) = 0.566433354765746647188050807325058683443823543741343518... . - _Vaclav Kotesovec_, Jan 02 2016
%e A025157 a(12) = 6 because we have 12 = 11+1 = 10+2 = 9+3 = 8+4 = 7+4+1.
%p A025157 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>
%p A025157       ceil(i*(i+3)/6), 0, b(n, i-1)+b(n-i, min(n-i, i-3))))
%p A025157     end:
%p A025157 a:= n-> b(n$2):
%p A025157 seq(a(n), n=0..70);  # _Alois P. Heinz_, Jan 26 2022
%t A025157 nn=50; CoefficientList[Series[Sum[x^(j(3j-1)/2)Product[1/(1-x^i), {i, 1, j}], {j, 0, nn}], {x, 0, nn}], x] (* _Geoffrey Critzer_, Jul 21 2013 *)
%o A025157 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=0, N, x^(k*(3*k-1)/2)/prod(j=1, k, 1-x^j))) \\ _Seiichi Manyama_, Jan 13 2022
%Y A025157 Cf. A003114, A096401, A263719.
%Y A025157 Column k=3 of A194543.
%K A025157 nonn
%O A025157 0,6
%A A025157 _Clark Kimberling_
%E A025157 Prepended a(0)=1, _Joerg Arndt_, Jul 21 2013