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A376077 Number of partitions of subsets of [n] containing n > 0, where consecutive integers are required to be in different parts.

%I A376077 #21 Sep 18 2024 07:41:17
%S A376077 1,1,2,6,19,68,269,1168,5516,28117,153668,895345,5534292,36137736,
%T A376077 248364343,1790801964,13508326353,106329846806,871423555238,
%U A376077 7420685528453,65539734707912,599363070599885,5666859173305898,55317197561841526,556788566486730535
%N A376077 Number of partitions of subsets of [n] containing n > 0, where consecutive integers are required to be in different parts.
%H A376077 Alois P. Heinz, <a href="/A376077/b376077.txt">Table of n, a(n) for n = 0..576</a>
%F A376077 a(0) = 1, a(n) = A261041(n) - A261041(n-1) for n>=1.
%F A376077 G.f.: Sum_{j>=0} A000110(j) * (x/(1-x^2))^j.
%e A376077 a(3) = 6: 3, 13, 1|3, 2|3, 13|2, 1|2|3.
%p A376077 b:= proc(n, m, i) option remember; `if`(n=0, 1, add(
%p A376077      `if`(i=j and j>0, 0, b(n-1, max(m, j), j)), j=0..m+1))
%p A376077     end:
%p A376077 a:= n-> b(n, 0$2)-`if`(n>0, b(n-1, 0$2), 0):
%p A376077 seq(a(n), n=0..30);
%t A376077 b[n_, m_, i_] := b[n, m, i] = If[n == 0, 1, Sum[If[i == j && j > 0, 0, b[n-1, Max[m, j], j]], {j, 0, m+1}]];
%t A376077 a[n_] := b[n, 0, 0] - If[n > 0, b[n-1, 0, 0], 0];
%t A376077 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Sep 18 2024, after _Alois P. Heinz_ *)
%Y A376077 Cf. A000045, A000110, A261041 (partial sums).
%K A376077 nonn
%O A376077 0,3
%A A376077 _Alois P. Heinz_, Sep 08 2024