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Number of partitions of n where each part i is marked with a word of length i over a binary alphabet whose letters appear in alphabetical order and both letters occur at least once in the partition.
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%I #12 Dec 08 2020 08:36:18

%S 3,12,40,104,279,654,1577,3560,8109,17734,39205,83996,181043,382856,

%T 811084,1694468,3545864,7340308,15205768,31259422,64253260,131314502,

%U 268332975,545854344,1110087515,2250051262,4558868119,9213241988,18613362500,37529700206

%N Number of partitions of n where each part i is marked with a word of length i over a binary alphabet whose letters appear in alphabetical order and both letters occur at least once in the partition.

%H Alois P. Heinz, <a href="/A293366/b293366.txt">Table of n, a(n) for n = 2..1000</a>

%F a(n) ~ c * 2^n, where c = A256155 = 18.563146563610114727475354232269284... - _Vaclav Kotesovec_, Oct 11 2017

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))

%p end:

%p a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):

%p seq(a(n), n=2..35);

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];

%t a[n_] := With[{k = 2}, Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];

%t a /@ Range[2, 35] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A261719.

%K nonn

%O 2,1

%A _Alois P. Heinz_, Oct 07 2017