A243923 Row sums of triangle A243920.

0, 1, 4, 12, 57, 469, 5409, 77321, 1304086, 25263208, 551790014, 13398776948, 357740951660, 10409057421898, 327640162774856, 11087710302096702, 401290657576717001, 15462394004585328685, 631795378164538352085, 27280160237622374011469, 1240933576265292837746859

Offset

0,3

Comments

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

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.

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

(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).

Links

Table of n, a(n) for n=0..20.

Example

Equals rows sums of triangle T = A243920, which begins:
0;
1, 0;
1, 3, 0;
4, 3, 5, 0;
27, 18, 5, 7, 0;
254, 159, 40, 7, 9, 0;
3048, 1836, 435, 70, 9, 11, 0;
44328, 26028, 5930, 903, 108, 11, 13, 0; ...
such that T as an infinite triangular matrix satisfies:
[I - T]^(-1) = Sum_{n>=0} T^n and equals T shifted up 1 row
(with all '1's replacing the main diagonal):
1;
1, 1;
4, 3, 1;
27, 18, 5, 1;
254, 159, 40, 7, 1;
3048, 1836, 435, 70, 9, 1;
44328, 26028, 5930, 903, 108, 11, 1; ...

Prog

(PARI) /* Get row sums using the g.f. for columns in triangle A243920: */
{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))}
{for(n=0, 20, print1(sum(k=0, n, A243920(n, k)), ", "))}

Crossrefs

Cf. A243920, A243921, A243922, A208678.

Sequence in context: A208940 A209068 A177265 * A192331 A068525 A067755

Adjacent sequences: A243920 A243921 A243922 * A243924 A243925 A243926

Keyword

nonn

Author

Paul D. Hanna, Jun 15 2014

Status

approved