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A264396 Number of partitions of n such that the part sizes occurring in it form an interval that does not start at 1.

%I A264396 #19 May 24 2018 09:29:13
%S A264396 0,1,1,2,2,3,3,4,5,5,6,8,8,9,12,12,14,17,18,21,25,26,30,36,39,43,51,
%T A264396 55,62,73,78,88,101,110,125,141,154,172,195,215,238,269,294,327,368,
%U A264396 402,446,498,547,606,672,737,814,903,991,1091,1205,1320,1452,1603,1752,1924,2118,2314,2539,2785,3042,3329,3648,3984
%N A264396 Number of partitions of n such that the part sizes occurring in it form an interval that does not start at 1.
%C A264396 The partitions in the definition are called non-complete gap-free (see the Grabner et al. reference).
%C A264396 a(n) = number of partitions of n where the largest part occurs at least twice and all other parts are distinct. Example: a(9) = 5 because we have 441, 333, 3321, 22221, and 111111111.
%H A264396 Alois P. Heinz, <a href="/A264396/b264396.txt">Table of n, a(n) for n = 1..10000</a>
%H A264396 P. J. Grabner, A. Knopfmacher, <a href="https://www.math.tugraz.at/fosp/pdfs/tugraz_0087.pdf">Analysis of some new partition statistics</a>, Ramanujan J., 12, 2006, 439-454.
%F A264396 G.f.: g = sum((x^{2j}/(1-x^j))*product(1+x^i, i=1..j), j=1..infinity).
%F A264396 a(n) ~ 3^(1/4) * Pi * exp(Pi*sqrt(n/3)) / (24 * n^(5/4)). - _Vaclav Kotesovec_, May 24 2018
%e A264396 a(9) = 5 because there are these partitions of 9: 9, 54, 432, 333, and 3222.
%p A264396 g := sum(x^(2*j)*(product(1+x^i, i = 1 .. j-1))/(1-x^j), j = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 1 .. 70);
%p A264396 # second Maple program:
%p A264396 b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, 0, 1),
%p A264396       `if`(i<1 or n<i, 0, add(b(n-i*j, i-1), j=1..n/i)))
%p A264396     end:
%p A264396 a:= n-> add(b(n, i), i=2..n):
%p A264396 seq(a(n), n=1..70);  # _Alois P. Heinz_, Nov 29 2015
%t A264396 b[n_, i_] := b[n, i] = If[n == 0, If[i == 0, 0, 1], If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; a[n_] := Sum[b[n, i], {i, 2, n}]; Table[a[n], {n, 1, 70}] (* _Jean-François Alcover_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y A264396 Cf. A251729.
%K A264396 nonn
%O A264396 1,4
%A A264396 _Emeric Deutsch_, Nov 17 2015