A073474 Triangle T(n,k) read by rows, where o.g.f. for T(n,k) is n!*Sum_{k=0..n} (1+x)^(n-k)/k!.
1, 2, 1, 5, 6, 2, 16, 33, 24, 6, 65, 196, 228, 120, 24, 326, 1305, 2120, 1740, 720, 120, 1957, 9786, 20550, 23160, 14760, 5040, 720, 13700, 82201, 212352, 305970, 265440, 138600, 40320, 5040, 109601, 767208, 2356424, 4146576, 4571280, 3232320, 1431360, 362880, 40320
Offset: 0
Examples
Triangle begins:
1;
2, 1;
5, 6, 2;
16, 33, 24, 6;
65, 196, 228, 120, 24;
326, 1305, 2120, 1740, 720, 120;
...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Programs
-
Maple
G:=simplify(series(exp(x)/(1-x-x*y),x=0,13)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(n!*coeff(G,x^n)) od: seq(seq(coeff(y*P[n],y^k),k=1..n+1),n=0..9); # second Maple program: b:= proc(n, k) option remember; `if`(k>n, 0, `if`(k=0, 1, n*(b(n-1, k-1)+b(n-1, k)))) end: T:= (n, k)-> b(n+1, k+1)/(n+1): seq(seq(T(n, k), k=0..n), n=0..9); # Alois P. Heinz, Sep 12 2019 -
Mathematica
b[n_, k_] := b[n, k] = If[k>n, 0, If[k==0, 1, n (b[n-1, k-1]+b[n-1, k])]]; T[n_, k_] := b[n+1, k+1]/(n+1); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2019, after Alois P. Heinz *) T[n_, k_] := Sum[Binomial[j, k] FactorialPower[n, j], {j, 0, n}]; (* Peter Luschny, Oct 16 2024 *) -
SageMath
def T(n, k): return sum(binomial(j, k) * falling_factorial(n, j) for j in range(n+1)) for n in range(8): print([T(n, k) for k in range(n+1)]) # Peter Luschny, Oct 16 2024
Formula
E.g.f.: exp(x)/(1-x-x*y). - Vladeta Jovovic, Oct 17 2003
T(n, k) = Sum_{j=0..n} binomial(j, k)*FallingFactorial(n, j). - Peter Luschny, Oct 16 2024
Extensions
Edited by Emeric Deutsch, Jun 10 2004
Comments