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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
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
Sequence in context: A208940 A209068 A177265 * A192331 A068525 A067755
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 15 2014
STATUS
approved

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Last modified April 21 09:27 EDT 2024. Contains 371851 sequences. (Running on oeis4.)