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
Examples
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 ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..465 (first 30 rows)
Programs
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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
Comments