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A046803 Triangle of numbers related to Eulerian numbers.

%I A046803 #17 Mar 08 2020 00:21:26
%S A046803 1,1,2,1,6,3,1,14,22,4,1,30,105,65,5,1,62,416,581,171,6,1,126,1491,
%T A046803 3920,2695,420,7,1,254,5034,22506,29310,11180,988,8,1,510,16365,
%U A046803 116667,256317,188361,43041,2259,9,1,1022,51892,564667,1945297,2419897,1090135
%N A046803 Triangle of numbers related to Eulerian numbers.
%D A046803 D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
%H A046803 Andrew Howroyd, <a href="/A046803/b046803.txt">Table of n, a(n) for n = 1..1275</a>
%F A046803 T(n, k) = Sum_{i=1..n} binomial(n,i) * A008292(n-i, k-1).
%F A046803 E.g.f.: exp(x*y)*(exp(x)-1)*(y-1)/(y*exp(x)-exp(x*y)). - _Vladeta Jovovic_, Sep 20 2003
%e A046803 Triangle begins
%e A046803   1;
%e A046803   1,   2;
%e A046803   1,   6,    3;
%e A046803   1,  14,   22,    4;
%e A046803   1,  30,  105,   65,    5;
%e A046803   1,  62,  416,  581,  171,   6;
%e A046803   1, 126, 1491, 3920, 2695, 420, 7;
%e A046803   ...
%t A046803 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_ *)
%o A046803 (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))))]}
%o A046803 { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Mar 07 2020
%o A046803 (PARI) \\ here U(n,k) is A008292.
%o A046803 U(n, k)={sum(j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j))};
%o A046803 T(n, k)={sum(i=1, n, binomial(n,i)*U(n-i, k-1))} \\ _Andrew Howroyd_, Mar 07 2020
%Y A046803 Row sums give A002627.
%Y A046803 Cf. A008292 (Eulerian numbers), A046802.
%K A046803 nonn,tabl,easy,nice
%O A046803 1,3
%A A046803 _N. J. A. Sloane_
%E A046803 More terms from _Vladeta Jovovic_, Sep 20 2003