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A157836 Triangle read by rows where T(n,k) is the number of factorizations of (n+1)! into k distinct factors.

%I A157836 #7 Feb 01 2020 23:34:29
%S A157836 1,1,1,1,3,1,1,7,7,1,1,14,28,13,1,1,29,103,95,24,1,1,47,273,448,249,
%T A157836 41,1,1,79,725,1897,1837,671,74,1,1,134,1876,7301,10856,6780,1686,127,
%U A157836 1,1,269,5791,31811,65782,59434,24017,3960,197,1,1,395,12061,92987,272932,362956,232152,69765,8703,323,1
%N A157836 Triangle read by rows where T(n,k) is the number of factorizations of (n+1)! into k distinct factors.
%C A157836 n-th row has n terms; first and last term in each row = 1.
%H A157836 Andrew Howroyd, <a href="/A157836/b157836.txt">Table of n, a(n) for n = 1..465</a> (first 30 rows)
%e A157836 Triangle begins:
%e A157836 2! 1
%e A157836 3! 1 1
%e A157836 4! 1 3 1
%e A157836 5! 1 7 7 1
%e A157836 6! 1 14 28 13 1
%e A157836 7! 1 29 103 95 24 1
%e A157836 8! 1 47 273 448 249 41 1
%e A157836 9! 1 79 725 1897 1837 671 74 1
%e A157836 10! 1 134 1876 7301 10856 6780 1686 127 1
%e A157836 11! 1 269 5791 31811 65782 59434 24017 3960 197 1
%e A157836 12! 1 395 12061 92987 272932 362956 232152 69765 8703 323 1
%e A157836 ...
%o A157836 (PARI)
%o A157836 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A157836 D(p, n, sig)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(r=EulerT(v)); prod(i=1, #sig, r[sig[i]])/prod(i=1, #v, i^v[i]*v[i]!)}
%o A157836 detail(sig)={my(m=vecsum(sig)+1,n=vecmax(sig), q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(y+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, sig), [1, n]); s*q[#q-j]*y^m)/(1+y))}
%o A157836 row(n)={if(n<=1, [], Vecrev(detail(factor(n!)[,2])))}
%o A157836 { for(n=1, 10, print(row(n+1))) } \\ _Andrew Howroyd_, Feb 01 2020
%Y A157836 A157612 gives row sums. A157672 gives 2nd column.
%K A157836 nonn,tabl
%O A157836 1,5
%A A157836 _Ray Chandler_, Mar 07 2009