A132623 Triangle T, read by rows, where 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, where T^n denotes the n-th matrix power of T.

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, 6098, 2992, 861, 188, 35, 6, 7, 0, 64050, 30800, 8589, 1788, 310, 48, 7, 8, 0, 759817, 360110, 98238, 19800, 3275, 474, 63, 8, 9, 0, 10028799, 4701734, 1262208, 248624

Offset

0,5

Links

Paul D. Hanna, Rows n = 0..45, flattened.

Formula

G.f. of column k: (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).

T(n,k) = [x^n] { (k+1)*x^(k+1) - Sum_{m=k+1..n-1} T(m,k)*x^m*(1-x)^(m-k) / Product_{j=k+1..m-1} (1+j*x) } for n>k with T(n,k)=0 when k>=n.

Example

Triangle 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;
6098, 2992, 861, 188, 35, 6, 7, 0;
64050, 30800, 8589, 1788, 310, 48, 7, 8, 0;
759817, 360110, 98238, 19800, 3275, 474, 63, 8, 9, 0; ...
-------------------------------------
MATRIX POWER SERIES PROPERTY.
[I - T]^-1 = Sum_{n>=0} T^n and equals T shifted up 1 row
(with '1's in the main diagonal):
1;
1, 1;
3, 2, 1;
14, 8, 3, 1;
87, 46, 15, 4, 1;
669, 338, 102, 24, 5, 1; ...
-------------------------------------
GENERATE T FROM MATRIX POWERS OF T.
Matrix square T^2 begins:
0;
0, 0;
2, 0, 0;
5, 6, 0, 0;
23, 14, 12, 0, 0;
143, 78, 27, 20, 0, 0; ...
so that T(4,1) = T(3,1) + [T^2](3,1) = 2 + 6 = 8;
and T(3,0) = T(2,0) + [T^2](2,0) = 1 + 2 = 3.
Matrix cube T^3 begins:
0;
0, 0;
0, 0, 0;
6, 0, 0, 0;
26, 24, 0, 0, 0;
165, 94, 60, 0, 0, 0; ...
so that T(5,1) = T(4,1) + [T^2](4,1) + [T^3](4,1) = 8 + 14 + 24 = 46;
and T(4,0) = T(3,0) + [T^2](3,0) + [T^3](3,0) = 3 + 5 + 6 = 14.
-------------------------------------
ILLUSTRATE G.F. FOR COLUMN k:
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) /* Using the matrix power formula: */
T(n, k)=local(M=if(n<=0, Mat(1), matrix(n, n, r, c, if(r>=c, T(r-1, c-1))))); if(n<k || k<0, 0, if(n==k, 0, if(n==k+1, n, sum(j=1, n-k-1, (M^j)[n, k+1]) )))
(PARI) /* Using the g.f. formula for columns: */
T(n, k)=if(n<k+1, 0, polcoeff((k+1)*x^(k+1)-sum(m=k+1, n-1, T(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, 15, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A132624 (column 0), A208676, A208677, A208678.

Sequence in context: A359674 A323248 A324397 * A277890 A243403 A051613

Adjacent sequences: A132620 A132621 A132622 * A132624 A132625 A132626

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 25 2007

Status

approved