VOOZH about

URL: https://oeis.org/A208678

⇱ A208678 - OEIS

login
A208678
Row sums of triangle A132623.
5
1, 3, 8, 29, 157, 1144, 10187, 105600, 1241794, 16287457, 235308853, 3708090433, 63234233743, 1159318599835, 22725352050303, 474059968069223, 10481049913889360, 244727123398669044, 6015958354315188049, 155261610128701766290, 4196413541685139001486
OFFSET
1,2
COMMENTS
Triangle T = A132623 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) = n+1 and T(n,n)=0 for n>=0.
Also, column k of triangle T = A132623 obeys the rule:
(k+1)*x^(k+1) = Sum_{n>=0} T(n,k) * x^n * (1-x)^(n-k) / Product_{j=k+1..n-1} (1+j*x).
EXAMPLE
Triangle A132623 begins:
0;
1, 0;
1, 2, 0;
3, 2, 3, 0;
14, 8, 3, 4, 0;
87, 46, 15, 4, 5, 0;
669, 338, 102, 24, 5, 6, 0; ...
for which this sequence equals the row sums.
MATRIX POWER SERIES PROPERTY OF T = A132623:
Let T = A132623, then [I - T]^-1 = Sum_{n>=0} T^n yields:
1;
1, 1;
3, 2, 1;
14, 8, 3, 1;
87, 46, 15, 4, 1;
669, 338, 102, 24, 5, 1; ...
which equals T shifted up 1 row (but with 1's in the main diagonal).
ILLUSTRATE G.F. FOR COLUMN k OF T = A132623:
k=0: x = T(1,0)*x*(1-x) + T(2,0)*x^2*(1-x)^2/((1+x)) + T(3,0)*x^3*(1-x)^3/((1+x)*(1+2*x)) + T(4,0)*x^4*(1-x)^4/((1+x)*(1+2*x)*(1+3*x)) +...
k=1: 2*x^2 = T(2,1)*x^2*(1-x) + T(3,1)*x^3*(1-x)^2/((1+2*x)) + T(4,1)*x^4*(1-x)^3/((1+2*x)*(1+3*x)) + T(5,1)*x^5*(1-x)^4/((1+2*x)*(1+3*x)*(1+4*x)) +...
k=2: 3*x^3 = T(3,2)*x^3*(1-x) + T(4,2)*x^4*(1-x)^2/((1+3*x)) + T(5,2)*x^5*(1-x)^3/((1+3*x)*(1+4*x)) + T(6,2)*x^6*(1-x)^4/((1+3*x)*(1+4*x)*(1+5*x)) +...
PROG
(PARI) /* Get row sums using the g.f. for columns in triangle A132623: */
{A132623(n, k)=if(n<k+1, 0, polcoeff((k+1)*x^(k+1)-sum(m=k+1, n-1, A132623(m, k)*x^m*(1-x)^(m-k)/prod(j=k+1, m-1, 1+j*x+x*O(x^n))), n))}
{for(n=0, 20, print1(sum(k=0, n, A132623(n, k)), ", "))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 14 2012
STATUS
approved