A046803 Triangle of numbers related to Eulerian numbers.

1, 1, 2, 1, 6, 3, 1, 14, 22, 4, 1, 30, 105, 65, 5, 1, 62, 416, 581, 171, 6, 1, 126, 1491, 3920, 2695, 420, 7, 1, 254, 5034, 22506, 29310, 11180, 988, 8, 1, 510, 16365, 116667, 256317, 188361, 43041, 2259, 9, 1, 1022, 51892, 564667, 1945297, 2419897, 1090135

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice

Examples

			Triangle begins
  1;
  1,   2;
  1,   6,    3;
  1,  14,   22,    4;
  1,  30,  105,   65,    5;
  1,  62,  416,  581,  171,   6;
  1, 126, 1491, 3920, 2695, 420, 7;
  ...
		

References

• D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Row sums give A002627.

Cf. A008292 (Eulerian numbers), A046802.

Programs

Mathematica

egf = Exp[x*y]*(Exp[x]-1)*((y-1)/(y*Exp[x] - Exp[x*y])); row[n_] := Last[ CoefficientList[ Series[egf, {x, 0, n}, {y, 0, n}], {x, y}]]*n!; Flatten[ Table[ row[n], {n, 1, 10}]] (* Jean-François Alcover, Dec 20 2012, after Vladeta Jovovic *)

PARI

T(n)={my(A=O(x*x^n)); [Vecrev(p) | p<-Vec(serlaplace(exp(x*y + A)*(exp(x + A)-1)*(y-1)/(y*exp(x + A)-exp(x*y + A))))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Mar 07 2020

PARI

\\ here U(n,k) is A008292.
U(n, k)={sum(j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j))};
T(n, k)={sum(i=1, n, binomial(n,i)*U(n-i, k-1))} \\ Andrew Howroyd, Mar 07 2020

Formula

T(n, k) = Sum_{i=1..n} binomial(n,i) * A008292(n-i, k-1).

E.g.f.: exp(x*y)*(exp(x)-1)*(y-1)/(y*exp(x)-exp(x*y)). - Vladeta Jovovic, Sep 20 2003

Extensions

More terms from Vladeta Jovovic, Sep 20 2003