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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. 2

%I #7 Feb 01 2020 23:34:29

%S 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 41,1,1,79,725,1897,1837,671,74,1,1,134,1876,7301,10856,6780,1686,127,

%U 1,1,269,5791,31811,65782,59434,24017,3960,197,1,1,395,12061,92987,272932,362956,232152,69765,8703,323,1

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

%C n-th row has n terms; first and last term in each row = 1.

%H Andrew Howroyd, <a href="/A157836/b157836.txt">Table of n, a(n) for n = 1..465</a> (first 30 rows)

%e Triangle begins:

%e 2! 1

%e 3! 1 1

%e 4! 1 3 1

%e 5! 1 7 7 1

%e 6! 1 14 28 13 1

%e 7! 1 29 103 95 24 1

%e 8! 1 47 273 448 249 41 1

%e 9! 1 79 725 1897 1837 671 74 1

%e 10! 1 134 1876 7301 10856 6780 1686 127 1

%e 11! 1 269 5791 31811 65782 59434 24017 3960 197 1

%e 12! 1 395 12061 92987 272932 362956 232152 69765 8703 323 1

%e ...

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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 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 row(n)={if(n<=1, [], Vecrev(detail(factor(n!)[,2])))}

%o { for(n=1, 10, print(row(n+1))) } \\ _Andrew Howroyd_, Feb 01 2020

%Y A157612 gives row sums. A157672 gives 2nd column.

%K nonn,tabl

%O 1,5

%A _Ray Chandler_, Mar 07 2009

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Last modified September 12 09:23 EDT 2024. Contains 375850 sequences. (Running on oeis4.)