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A179927 Triangle of centered orthotopic numbers.

%I A179927 #9 Jul 22 2025 08:24:02
%S A179927 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,
%T A179927 65,275,337,189,61,13,2,1,129,793,1267,881,341,85,15,2
%N A179927 Triangle of centered orthotopic numbers.
%C A179927 T(n, k) = [x^k] series[ H(n - k, x) ]
%C A179927 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.
%H A179927 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FigurateNumber">Figurate number - a very short introduction</a>
%e A179927 1
%e A179927 1, 2
%e A179927 1, 3, 2
%e A179927 1, 5, 5, 2
%e A179927 1, 9, 13, 7, 2
%e A179927 1, 17, 35, 25, 9, 2
%e A179927 1, 33, 97, 91, 41, 11, 2
%p A179927 E := (n,x) -> `if`(n=0,1,x*(1-x)*diff(E(n-1,x),x)+E(n-1,x)*(1+(n-1)*x));
%p A179927 H := (n,x) -> E(n,x)*(1+x)/(1-x)^(n+1);
%p A179927 A179927 := (n,k) -> coeff(series(H(n-k,x),x,18),x,k);
%p A179927 seq(print(seq(A179927(n,k),k=0..n)),n=0..6);
%t A179927 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);
%t A179927 h[n_, x_] := e[n, x] (1+x)/(1-x)^(n+1);
%t A179927 T[n_, k_] := SeriesCoefficient[h[n-k, x], {x, 0, k}];
%t A179927 Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Jean-François Alcover_, Jun 17 2019, from Maple *)
%Y A179927 Cf. Row sums in A179928, triangle of orthotopic numbers is A009998.
%K A179927 nonn,tabl
%O A179927 0,3
%A A179927 _Peter Luschny_, Aug 02 2010