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

1,3

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

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

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

Row sums give A002627.

Cf. A008292 (Eulerian numbers), A046802.

Sequence in context: A034898 A059300 A321331 * A280789 A121468 A168151

Adjacent sequences: A046800 A046801 A046802 * A046804 A046805 A046806

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Sep 20 2003

Status

approved