|
%I #70 Nov 09 2020 11:38:15
%S 1,1,1,8,3,1,6,35,6,1,144,110,95,10,1,480,1594,585,205,15,1,5760,8064,
%T 8974,1995,385,21,1,5040,125292,70252,35329,5320,658,28,1,524160,
%U 684144,1178540,392364,110649,12096,1050,36,1,2177280,14215536,10683180,7260560,1630125,295113,24570,1590,45,1
%N 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.
%C Also the Bell transform of A265024. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%H Seiichi Manyama, <a href="/A075525/b075525.txt">Rows n = 1..100, flattened</a>
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F Row sums give n!*A000009(n).
%F From _Seiichi Manyama_, Nov 08 2020: (Start)
%F E.g.f.: exp(Sum_{n>0} u*A000593(n)*t^n/n).
%F 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)
%F 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.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 8, 3, 1;
%e 6, 35, 6, 1;
%e 144, 110, 95, 10, 1;
%e 480, 1594, 585, 205, 15, 1;
%e 5760, 8064, 8974, 1995, 385, 21, 1;
%e 5040, 125292, 70252, 35329, 5320, 658, 28, 1;
%e ...
%e 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.
%p # Adds (1,0,0,0,...) as row 0.
%p 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
%t T[n_, k_] := n! SeriesCoefficient[(Times @@ (1 + t^Range[n]))^u, {t, 0, n}, {u, 0, k}];
%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 04 2019 *)
%o (Sage) # uses[bell_matrix from A264428]
%o # Adds (1,0,0,0,..) as row 0.
%o d = lambda n: sum((-1)^(d+1)*n/d for d in divisors(n))
%o bell_matrix(lambda n: factorial(n)*d(n+1), 9) # _Peter Luschny_, Jan 26 2016
%o (PARI) a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)*n/d));
%o 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_
%Y Column k=1..3 give A265024, A338787, A338788.
%Y Cf. A000593, A008298.
%K nonn,tabl
%O 1,4
%A _Vladeta Jovovic_, Oct 11 2002
|
