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

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, 41, 1, 1, 79, 725, 1897, 1837, 671, 74, 1, 1, 134, 1876, 7301, 10856, 6780, 1686, 127, 1, 1, 269, 5791, 31811, 65782, 59434, 24017, 3960, 197, 1, 1, 395, 12061, 92987, 272932, 362956, 232152, 69765, 8703, 323, 1

Offset

1,5

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..465 (first 30 rows)

Example

Triangle begins:
2! 1
3! 1 1
4! 1 3 1
5! 1 7 7 1
6! 1 14 28 13 1
7! 1 29 103 95 24 1
8! 1 47 273 448 249 41 1
9! 1 79 725 1897 1837 671 74 1
10! 1 134 1876 7301 10856 6780 1686 127 1
11! 1 269 5791 31811 65782 59434 24017 3960 197 1
12! 1 395 12061 92987 272932 362956 232152 69765 8703 323 1
...

Prog

(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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]!)}
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))}
row(n)={if(n<=1, [], Vecrev(detail(factor(n!)[, 2])))}
{ for(n=1, 10, print(row(n+1))) } \\ Andrew Howroyd, Feb 01 2020

Crossrefs

A157612 gives row sums. A157672 gives 2nd column.

Sequence in context: A082039 A367505 A176331 * A205497 A063394 A344527

Adjacent sequences: A157833 A157834 A157835 * A157837 A157838 A157839

Keyword

nonn,tabl

Author

Ray Chandler, Mar 07 2009

Status

approved