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A261732 Number of partitions of 2n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur at least once in the partition.

%I A261732 #15 Dec 29 2020 09:03:18
%S A261732 1,2,40,1751,136516,16993932,3112631737,792794568624,269047552566188,
%T A261732 117558189248146187,64343348623810658670,43136101281780352785381,
%U A261732 34769493663713954019980281,33175795620329111188048543481,36981665738825428991431755534146
%N A261732 Number of partitions of 2n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur at least once in the partition.
%H A261732 Alois P. Heinz, <a href="/A261732/b261732.txt">Table of n, a(n) for n = 0..200</a>
%F A261732 a(n) = A261719(2n,n).
%F A261732 a(n) ~ c * d^n * n! * (n-1)!, where d = -4/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 6.1765546094834803582316801640508765536... and c = 0.52251062602313321387485... . - _Vaclav Kotesovec_, Feb 18 2017
%e A261732 a(0) = 1: the empty partition.
%e A261732 a(1) = 2: 2aa, 1a1a.
%e A261732 a(2) = 40: 4aaab, 4aabb, 4abbb, 3aaa1b, 3aab1a, 3aab1b, 3abb1a, 3abb1b, 3bbb1a, 2aa2ab, 2aa2bb, 2ab2aa, 2ab2ab, 2ab2bb, 2bb2aa, 2bb2ab, 2aa1a1b, 2aa1b1a, 2aa1b1b, 2ab1a1a, 2ab1a1b, 2ab1b1a, 2ab1b1b, 2bb1a1a, 2bb1a1b, 2bb1b1a, 1a1a1a1b, 1a1a1b1a, 1a1a1b1b, 1a1b1a1a, 1a1b1a1b, 1a1b1b1a, 1a1b1b1b, 1b1a1a1a, 1b1a1a1b, 1b1a1b1a, 1b1a1b1b, 1b1b1a1a, 1b1b1a1b, 1b1b1b1a.
%p A261732 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261732       b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
%p A261732     end:
%p A261732 a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A261732 seq(a(n), n=0..15);
%t A261732 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 A261732 a[n_] := Sum[b[2n, 2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
%t A261732 a /@ Range[0, 15] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A261732 Cf. A261719, A226775.
%K A261732 nonn
%O A261732 0,2
%A A261732 _Alois P. Heinz_, Aug 30 2015