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

Author

Peter Luschny, Aug 02 2010

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• Peter Luschny, Figurate number - a very short introduction

Crossrefs

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

Programs

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 *)