A287315 - OEIS

%I #13 Jan 10 2018 20:33:26

%S 1,0,1,0,1,3,0,1,16,19,0,1,65,299,211,0,1,246,3156,7346,3651,0,1,917,

%T 28722,160322,237517,90921,0,1,3424,245407,2864912,9302567,9903776,

%U 3081513,0,1,12861,2041965,46261609,288196659,632274183,520507423,136407699

%N Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287316.

%F Sum_{k=0..n} T(n,k) = A001044(n).

%e Triangle starts:

%e 0: [1]

%e 1: [0, 1]

%e 2: [0, 1,    3]

%e 3: [0, 1,   16,     19]

%e 4: [0, 1,   65,    299,     211]

%e 5: [0, 1,  246,   3156,    7346,    3651]

%e 6: [0, 1,  917,  28722,  160322,  237517,   90921]

%e 7: [0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513]

%e ...

%e Let q4(x) = (x + 65*x^2 + 299*x^3 + 211*x^4) / (1-x)^5 then the coefficients of the series expansion of q4 give A169712, which is column 4 of A287316.

%p Delta := proc(a, n) local del, A, u;

%p A := [seq(a(j), j=0..n+1)]; del := (a, k) -> `if`(k=0, a(0), a(k)-a(k-1));

%p for u from 0 to n do A := [seq(del(k -> A[k+1], j), j=0..n)] od end:

%p A287315_row := n -> Delta(A287314_poly(n), n):

%p for n from 0 to 7 do A287315_row(n) od;

%p A287315_eulerian := (n,x) -> add(A287315_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):

%p for n from 0 to 4 do A287315_eulerian(n,x) od;

%Y T(n,n) = A000275(n).

%Y Cf. A192721 (variant), A001044, A287314, A287316.

%K nonn,tabl

%O 0,6

%A _Peter Luschny_, May 29 2017