A075525 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*t^n/n! = ((1+t)*(1+t^2)*(1+t^3)...)^u.
1, 1, 1, 8, 3, 1, 6, 35, 6, 1, 144, 110, 95, 10, 1, 480, 1594, 585, 205, 15, 1, 5760, 8064, 8974, 1995, 385, 21, 1, 5040, 125292, 70252, 35329, 5320, 658, 28, 1, 524160, 684144, 1178540, 392364, 110649, 12096, 1050, 36, 1, 2177280, 14215536, 10683180, 7260560, 1630125, 295113, 24570, 1590, 45, 1
Offset: 1
Examples
Triangle begins:
1;
1, 1;
8, 3, 1;
6, 35, 6, 1;
144, 110, 95, 10, 1;
480, 1594, 585, 205, 15, 1;
5760, 8064, 8974, 1995, 385, 21, 1;
5040, 125292, 70252, 35329, 5320, 658, 28, 1;
...
exp(Sum_{n>0} u*A000593(n)*t^n/n) = 1 + u*t/1! + (u+u^2)*t^2/2! + (8*u+3*u^2+u^3)*t^3/3! + (6*u+35*u^2+6*u^3+u^4)*t^4/4! + ... - _Seiichi Manyama_, Nov 08 2020.
Links
- Seiichi Manyama, Rows n = 1..100, flattened
- Peter Luschny, The Bell transform.
Programs
-
Maple
# Adds (1,0,0,0,...) as row 0. seq(PolynomialTools[CoefficientList](n!*coeff(series(mul((1+z^k)^u, k=1..20),z,20),z,n),u), n=0..9); # Peter Luschny, Jan 26 2016
-
Mathematica
T[n_, k_] := n! SeriesCoefficient[(Times @@ (1 + t^Range[n]))^u, {t, 0, n}, {u, 0, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2019 *) -
PARI
a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)*n/d)); T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1))) \\ Seiichi Manyama, Nov 08 2020 after Peter Luschny
-
Sage
# uses[bell_matrix from A264428] # Adds (1,0,0,0,..) as row 0. d = lambda n: sum((-1)^(d+1)*n/d for d in divisors(n)) bell_matrix(lambda n: factorial(n)*d(n+1), 9) # Peter Luschny, Jan 26 2016
Formula
Row sums give n!*A000009(n).
From Seiichi Manyama, Nov 08 2020: (Start)
E.g.f.: exp(Sum_{n>0} u*A000593(n)*t^n/n).
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} A000593(k)*T(n-k; u)/(n-k)!, T(0; u) = 1. (End)
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} A000593(i_j)/i_j. - Seiichi Manyama, Nov 09 2020.
Comments