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A243923 Row sums of triangle A243920. 3

%I #9 Jun 15 2014 22:39:32

%S 0,1,4,12,57,469,5409,77321,1304086,25263208,551790014,13398776948,

%T 357740951660,10409057421898,327640162774856,11087710302096702,

%U 401290657576717001,15462394004585328685,631795378164538352085,27280160237622374011469,1240933576265292837746859

%N Row sums of triangle A243920.

%C Triangle T = A243920 is generated by sums of matrix powers of itself such that:

%C T(n,k) = Sum_{j=1..n-k-1} [T^j](n-1,k) with T(n+1,n) = 2*n+1 and T(n,n)=0 for n>=0.

%C Also, column k of triangle T = A243920 obeys the rule:

%C (2*k+1)*x^(k+1) = Sum_{n>=0} T(n,k) * x^n * (1-x)^(n-k) / Product_{j=k+1..n-1} (1+2*j*x).

%e Equals rows sums of triangle T = A243920, which begins:

%e 0;

%e 1, 0;

%e 1, 3, 0;

%e 4, 3, 5, 0;

%e 27, 18, 5, 7, 0;

%e 254, 159, 40, 7, 9, 0;

%e 3048, 1836, 435, 70, 9, 11, 0;

%e 44328, 26028, 5930, 903, 108, 11, 13, 0; ...

%e such that T as an infinite triangular matrix satisfies:

%e [I - T]^(-1) = Sum_{n>=0} T^n and equals T shifted up 1 row

%e (with all '1's replacing the main diagonal):

%e 1;

%e 1, 1;

%e 4, 3, 1;

%e 27, 18, 5, 1;

%e 254, 159, 40, 7, 1;

%e 3048, 1836, 435, 70, 9, 1;

%e 44328, 26028, 5930, 903, 108, 11, 1; ...

%o (PARI) /* Get row sums using the g.f. for columns in triangle A243920: */

%o {A243920(n, k)=if(n<k+1, 0, polcoeff((2*k+1)*x^(k+1)-sum(m=k+1, n-1, A243920(m, k)*x^m*(1-x)^(m-k)/prod(j=k+1, m-1, 1+2*j*x+x*O(x^n))), n))}

%o {for(n=0, 20, print1(sum(k=0, n, A243920(n, k)), ", "))}

%Y Cf. A243920, A243921, A243922, A208678.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 15 2014

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