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Triangle T(n,k) = coefficient of x^n*y^k/(n!*k!) in 1/(1-x-y-x*y), read by rows in order 00, 10, 01, 20, 11, 02, ...
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%I #10 Aug 19 2024 14:59:59

%S 1,1,1,2,3,2,6,10,10,6,24,42,52,42,24,120,216,300,300,216,120,720,

%T 1320,1968,2268,1968,1320,720,5040,9360,14640,18576,18576,14640,9360,

%U 5040,40320,75600,122400,166320,184896,166320,122400,75600,40320

%N Triangle T(n,k) = coefficient of x^n*y^k/(n!*k!) in 1/(1-x-y-x*y), read by rows in order 00, 10, 01, 20, 11, 02, ...

%F E.g.f.: 1/(1-x-y-x*y).

%F T(n, k) = n!*2^k*Hypergeometric2F1([-k, -k], [-n], 1/2). - _Detlef Meya_, Aug 18 2024

%e Triangle begins:

%e 1;

%e 1,1;

%e 2,3,2;

%e 6,10,10,6;

%e ...

%p read transforms; SERIES2(1/(1-x-y-x*y),x,y,12): SERIES2TOLISTMULT(%,x,y,12);

%t T[n_, k_] := n!*2^k*Hypergeometric2F1[-k, -k, -n, 1/2]; Table[T[n,k], {n, 0, 8}, {k, 0, n}]//Flatten (* _Detlef Meya_, Aug 18 2024 *)

%Y Cf. A008288.

%K nonn,tabl,easy

%O 0,4

%A _N. J. A. Sloane_, Feb 02 2001