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 A175012 Triangle generated from the g.f of A000712 (i.e., 1/(1-x^m)^2) interleaved with zeros. 1

%I

%S 1,2,2,3,2,4,4,2,4,9,5,2,4,10,14,6,2,4,10,19,23,7,2,4,10,20,34,32,8,2,

%T 4,10,20,39,55,46,9,2,4,10,20,40,66,88,60,10

%N Triangle generated from the g.f of A000712 (i.e., 1/(1-x^m)^2) interleaved with zeros.

%C Row sums of the triangle = A000712.

%F Given 1/(1-x^m)^2 = S(x) = (1 + 2x + 3x^2 + ...), let a = S(x), b = S(x^2) (i.e., S(x) interleaved with one zero); S(x^3) = S(x) interleaved with two zeros = c, etc.; then row 1 = a, row 2 = a*b, row 3 = a*b*c, ...

%F Take finite differences of the array from the top down, becoming rows of the triangle.

%e First few rows of the array =

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

%e 1, 2, 5, 8, 14, 20, 30, 40, 55, 70, ...

%e 1, 2, 5, 10, 18, 30, 49, 74, 110, 158, ...

%e 1, 2, 5, 10, 20, 34, 59, 94, 149, 224, ...

%e 1, 2, 5, 10, 20, 36, 63, 104, 169, 264, ...

%e 1, 2, 5, 10, 20, 36, 65, 108, 179, 284, ...

%e ...

%e First few rows of the triangle =

%e 1;

%e 2;

%e 2, 3;

%e 2, 4, 4;

%e 2, 4, 9, 5;

%e 2, 4, 10, 14, 6;

%e 2, 4, 10, 19, 23, 7;

%e 2, 4, 10, 20, 34, 32, 8;

%e 2, 4, 10, 20, 39, 55, 46, 9;

%e 2, 4, 10, 20, 40, 66, 88, 60, 10;

%e ...

%Y Cf. A000712.

%K nonn,tabl,more

%O 0,2

%A _Gary W. Adamson_, Apr 03 2010

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Last modified September 20 17:42 EDT 2021. Contains 347588 sequences. (Running on oeis4.)