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%I #58 Oct 27 2023 20:57:18
%S 1,1,3,9,37,169,981,6429,49669,430861,4208925,45345165,536229373,
%T 6884917597,95473049469,1420609412637,22580588347741,381713065286173,
%U 6837950790434781,129378941557961565,2578133190722896861,53965646957320869469,1183822028149936497501
%N Sum of products of factorials of parts in all partitions of n.
%C Row sums of arrays A069123 and A134133. Row sums of triangle A134134.
%H Alois P. Heinz, <a href="/A077365/b077365.txt">Table of n, a(n) for n = 0..450</a> (terms n = 0..70 from Vincenzo Librandi)
%H J.-P. Bultel, A, Chouria, J.-G. Luque and O. Mallet, <a href="https://hal.archives-ouvertes.fr/hal-00793788">Word symmetric functions and the Redfield-Polya theorem</a>, 2013.
%F G.f.: 1/Product_{m>0} (1-m!*x^m).
%F Recurrence: a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(k/d).
%F a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 67/n^4 + 457/n^5 + 3734/n^6 + 35741/n^7 + 392875/n^8 + 4886114/n^9 + 67924417/n^10), for coefficients see A256125. - _Vaclav Kotesovec_, Mar 14 2015
%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j!)^k*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 18 2018
%e The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products of factorials of parts are 24,6,4,2,1 and their sum is a(4) = 37.
%e 1 + x + 3 x^2 + 9 x^3 + 37 x^4 + 169 x^5 + 981 x^6 + 6429 x^7 + 49669 x^8 + ...
%p b:= proc(n, i, j) option remember;
%p `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
%p `if`(i>n, 0, j^i*b(n-i, i, j+1))))
%p end:
%p a:= n-> b(n$2, 1):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 03 2013
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+`if`(i>n, 0, b(n-i, i)*i!)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 11 2016
%t Table[Plus @@ Map[Times @@ (#!) &, IntegerPartitions[n]], {n, 0, 20}] (* _Olivier Gérard_, Oct 22 2011 *)
%t a[ n_] := If[ n < 0, 0, Plus @@ Times @@@ (IntegerPartitions[ n] !)] (* _Michael Somos_, Feb 09 2012 *)
%t nmax=20; CoefficientList[Series[Product[1/(1-k!*x^k),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Mar 14 2015 *)
%t b[n_, i_, j_] := b[n, i, j] = If[n==0, 1, If[i<1, 0, b[n, i-1, j] + If[i>n, 0, j^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 12 2015, after _Alois P. Heinz_ *)
%o (PARI)
%o N=66; q='q+O('q^N);
%o gf= 1/prod(n=1,N, (1-n!*q^n) );
%o Vec(gf)
%o /* _Joerg Arndt_, Oct 06 2012 */
%Y Cf. A006906, A074141, A256125, A265950.
%Y Cf. A069123, A134133, A134134.
%Y Cf. A051296 (with compositions instead of partitions).
%K nonn
%O 0,3
%A _Vladeta Jovovic_, Nov 30 2002
%E Unnecessarily complicated mma code deleted by _N. J. A. Sloane_, Sep 21 2009
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