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URL: https://oeis.org/A046803

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A046803
Triangle of numbers related to Eulerian numbers.
1
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
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
KEYWORD
nonn,tabl,easy,nice
EXTENSIONS
More terms from Vladeta Jovovic, Sep 20 2003
STATUS
approved