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A243923 Row sums of triangle A243920. 3
0, 1, 4, 12, 57, 469, 5409, 77321, 1304086, 25263208, 551790014, 13398776948, 357740951660, 10409057421898, 327640162774856, 11087710302096702, 401290657576717001, 15462394004585328685, 631795378164538352085, 27280160237622374011469, 1240933576265292837746859 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 18 18:06 EDT 2022. Contains 353823 sequences. (Running on oeis4.)