A179927 Triangle of centered orthotopic numbers

1, 1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 9, 13, 7, 2, 1, 17, 35, 25, 9, 2, 1, 33, 97, 91, 41, 11, 2, 1, 65, 275, 337, 189, 61, 13, 2, 1, 129, 793, 1267, 881, 341, 85, 15, 2

Offset

0,3

Comments

T(n, k) = [x^k] series[ H(n - k, x) ]

Here H(n,x) = E(n,x)*(1+x)/(1-x)^(n+1) where E(n,x) are the Eulerian polynomials, E(0,x) = 1 and E(n,x) = sum_{k=0^{n-1}} W_{n,k} x^k for n > 0. W_{n,k} as in DLMF Table 26.14.1.

Links

Table of n, a(n) for n=0..44.

Peter Luschny, Figurate number - a very short introduction

Example

1
1, 2
1, 3, 2
1, 5, 5, 2
1, 9, 13, 7, 2
1, 17, 35, 25, 9, 2
1, 33, 97, 91, 41, 11, 2

Maple

E := (n, x) -> `if`(n=0, 1, x*(1-x)*diff(E(n-1, x), x)+E(n-1, x)*(1+(n-1)*x));
H := (n, x) -> E(n, x)*(1+x)/(1-x)^(n+1);
A179927 := (n, k) -> coeff(series(H(n-k, x), x, 18), x, k);
seq(print(seq(A179927(n, k), k=0..n)), n=0..6);

Mathematica

e[0, _] = 1; e[n_, x_] := e[n, x] = x(1-x) D[e[n-1, x], x] + e[n-1, x] (1 + (n-1)x);
h[n_, x_] := e[n, x] (1+x)/(1-x)^(n+1);
T[n_, k_] := SeriesCoefficient[h[n-k, x], {x, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 17 2019, from Maple *)

Crossrefs

Cf. Row sums in A179928, triangle of orthotopic numbers is A009998.

Sequence in context: A038497 A091355 A131245 * A104446 A131345 A134423

Adjacent sequences: A179924 A179925 A179926 * A179928 A179929 A179930

Keyword

nonn,tabl

Author

Peter Luschny, Aug 02 2010

Status

approved