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A right-skewed Pascal triangle, with interspersed 1's on main diagonal.
3

%I #11 Jun 12 2024 06:56:12

%S 1,1,1,1,0,1,1,0,2,1,1,0,3,0,1,1,0,4,0,3,1,1,0,5,0,6,0,1,1,0,6,0,10,0,

%T 4,1,1,0,7,0,15,0,10,0,1,1,0,8,0,21,0,20,0,5,1,1,0,9,0,28,0,35,0,15,0,

%U 1,1,0,10,0,36,0,56,0,35,0,6,1

%N A right-skewed Pascal triangle, with interspersed 1's on main diagonal.

%C Linked to Fibonacci numbers by row sums A127968. Inverse A127969 is linked to Genocchi numbers by its first column.

%F Column k is generated by x^(-2)*(x*(1-x)^(-1/2))^(k+2), where non-integer values are set to 0; or, column k has g.f. x^k/(1-x)^(1+k/2) if k is even, x^k if k is odd.

%e Triangle begins

%e 1;

%e 1, 1;

%e 1, 0, 1;

%e 1, 0, 2, 1;

%e 1, 0, 3, 0, 1;

%e 1, 0, 4, 0, 3, 1;

%e 1, 0, 5, 0, 6, 0, 1;

%e 1, 0, 6, 0, 10, 0, 4, 1;

%e 1, 0, 7, 0, 15, 0, 10, 0, 1;

%e 1, 0, 8, 0, 21, 0, 20, 0, 5, 1;

%e 1, 0, 9, 0, 28, 0, 35, 0, 15, 0, 1;

%e 1, 0, 10, 0, 36, 0, 56, 0, 35, 0, 6, 1;

%e 1, 0, 11, 0, 45, 0, 84, 0, 70, 0, 21, 0, 1;

%Y Cf. A127968, A127969.

%K easy,nonn,tabl

%O 0,9

%A _Paul Barry_, Feb 09 2007