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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
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 (list; table; graph; refs; listen; history; text; internal format)
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: A132733 A082039 A176331 * A205497 A063394 A193871

Adjacent sequences:  A157833 A157834 A157835 * A157837 A157838 A157839

KEYWORD

nonn,tabl

AUTHOR

Ray Chandler, Mar 07 2009

STATUS

approved

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Last modified January 17 16:01 EST 2021. Contains 340245 sequences. (Running on oeis4.)