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

1,4

Comments

Also the Bell transform of A265024. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Links

Seiichi Manyama, Rows n = 1..100, flattened

Peter Luschny, The Bell transform.

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.

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Column k=1..3 give A265024, A338787, A338788.

Cf. A000593, A008298.

Sequence in context: A092555 A001061 A259073 * A242048 A097890 A088453

Adjacent sequences: A075522 A075523 A075524 * A075526 A075527 A075528

Keyword

nonn,tabl

Author

Vladeta Jovovic, Oct 11 2002

Status

approved