A208678 Row sums of triangle A132623.

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).

Links

Table of n, a(n) for n=1..21.

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

Cf. A132623, A132623, A208676, A208677.

Sequence in context: A275166 A182117 A207828 * A162054 A289486 A245361

Adjacent sequences: A208675 A208676 A208677 * A208679 A208680 A208681

Keyword

nonn

Author

Paul D. Hanna, Mar 14 2012

Status

approved