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A179927 Triangle of centered orthotopic numbers 4

%I #5 Jun 17 2019 16:03:27

%S 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 65,275,337,189,61,13,2,1,129,793,1267,881,341,85,15,2

%N Triangle of centered orthotopic numbers

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

%C 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 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FigurateNumber"> Figurate number - a very short introduction </a>

%e 1

%e 1, 2

%e 1, 3, 2

%e 1, 5, 5, 2

%e 1, 9, 13, 7, 2

%e 1, 17, 35, 25, 9, 2

%e 1, 33, 97, 91, 41, 11, 2

%p 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 H := (n,x) -> E(n,x)*(1+x)/(1-x)^(n+1);

%p A179927 := (n,k) -> coeff(series(H(n-k,x),x,18),x,k);

%p seq(print(seq(A179927(n,k),k=0..n)),n=0..6);

%t 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 h[n_, x_] := e[n, x] (1+x)/(1-x)^(n+1);

%t T[n_, k_] := SeriesCoefficient[h[n-k, x], {x, 0, k}];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Jean-François Alcover_, Jun 17 2019, from Maple *)

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

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, Aug 02 2010

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)