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A274762 Number of sequences with up to n copies each of 1,2,...,n.

%I A274762 #22 May 24 2020 11:33:48
%S A274762 1,2,19,5248,191448941,1856296498826906,7843008902239185171370147,
%T A274762 21408941228439913825832318523364743824,
%U A274762 52400635808473472283994952631626957015306076632624953,152306240915343870544748050434914720360496623911547121447677238156864610
%N A274762 Number of sequences with up to n copies each of 1,2,...,n.
%H A274762 Alois P. Heinz, <a href="/A274762/b274762.txt">Table of n, a(n) for n = 0..26</a>
%F A274762 a(n) ~ exp(11/12) * n^(n^2 - n/2 + 1) / (2*Pi)^((n-1)/2). - _Vaclav Kotesovec_, May 24 2020
%e A274762 a(0) = 1: () = the empty sequence.
%e A274762 a(1) = 2: (), 1.
%e A274762 a(2) = 19: (), 1, 2, 11, 12, 21, 22, 112, 121, 122, 211, 212, 221, 1122, 1212, 1221, 2112, 2121, 2211.
%p A274762 b:= proc(n, k, i) option remember; `if`(k=0, 1,
%p A274762      `if`(i<1, 0, add(b(n, k-j, i-1)/j!, j=0..min(k, n))))
%p A274762     end:
%p A274762 a:= n-> add(b(n, k, n)*k!, k=0..n^2):
%p A274762 seq(a(n), n=0..10);
%t A274762 Table[Sum[k!*SeriesCoefficient[Sum[x^j/j!, {j, 0, n}]^n, {x, 0, k}], {k, 0, n^2}], {n, 0, 10}] (* _Vaclav Kotesovec_, May 24 2020 *)
%o A274762 (PARI) {a(n) = sum(i=0, n^2, i!*polcoef(sum(j=0, n, x^j/j!)^n, i))} \\ _Seiichi Manyama_, May 19 2019
%Y A274762 Row sums of A234574.
%Y A274762 Main diagonal of A308292.
%Y A274762 Cf. A000312, A034841.
%K A274762 nonn
%O A274762 0,2
%A A274762 _Alois P. Heinz_, Jul 04 2016