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A129481 a(n) = coefficient of x^n in n!*Product_{k=0..n} [Sum_{j=0..k} x^j/j! ].

%I A129481 #18 Feb 13 2024 03:35:56
%S A129481 1,1,3,19,175,2111,31321,550810,11194177,258068893,6653230111,
%T A129481 189653427206,5922604033567,201075967613262,7373834652641003,
%U A129481 290474615891145106,12232735359488840833,548429151685677131389
%N A129481 a(n) = coefficient of x^n in n!*Product_{k=0..n} [Sum_{j=0..k} x^j/j! ].
%C A129481 a(n) is also the number of ordered submultisets of A000707. - _J. M. Bergot_, Aug 13 2016
%H A129481 Vaclav Kotesovec, <a href="/A129481/b129481.txt">Table of n, a(n) for n = 0..150</a>
%F A129481 a(n) ~ c * n^n, where c = 0.660942456683588459181273625114230472913... . - _Vaclav Kotesovec_, Feb 10 2015
%e A129481 a(2) = [x^2] 2!*(1)*(1+x)*(1+x+x^2/2!) = [x^2] (2 +4*x +3*x^2 +x^3) = 3.
%e A129481 a(3) = [x^3] 3!*(1)*(1+x)*(1 + x + x^2/2!)*(1 + x + x^2/2! + x^3/3!) =
%e A129481 [x^3] (6 + 18*x + 24*x^2 + 19*x^3 +...) = 19.
%t A129481 Flatten[{1,Table[Coefficient[Expand[n!*Product[Sum[x^j/j!,{j,0,k}],{k,0,n}]],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 10 2015 *)
%o A129481 (PARI) {a(n)=n!*polcoeff(prod(k=0,n,sum(j=0,k,x^j/j!)+x*O(x^n)),n)}
%o A129481 (Magma)
%o A129481 m:=30; R<x>:=PowerSeriesRing(Integers(), m+2);
%o A129481 p:= func< n,x | (&*[ (&+[x^j/Factorial(j): j in [0..k]]) : k in [0..n]]) >;
%o A129481 A129481:= func< n | Coefficient(R!(Laplace( p(n,x) )), n) >;
%o A129481 [A129481(n): n in [0..m]]; // _G. C. Greubel_, Feb 12 2024
%o A129481 (SageMath)
%o A129481 def p(n,x): return product(sum(x^j/factorial(j) for j in range(k+1)) for k in range(n+1))
%o A129481 def A129481(n): return factorial(n)*( p(n,x) ).series(x, 101).list()[n]
%o A129481 [A129481(n) for n in range(31)] # _G. C. Greubel_, Feb 13 2024
%Y A129481 Cf. A000707.
%K A129481 nonn
%O A129481 0,3
%A A129481 _Paul D. Hanna_, Apr 17 2007