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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.
6
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
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; ...
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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.
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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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Aug 25 2007
STATUS
approved