A132623 - OEIS

%I #15 Jun 14 2017 00:35:31

%S 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,

%T 6098,2992,861,188,35,6,7,0,64050,30800,8589,1788,310,48,7,8,0,759817,

%U 360110,98238,19800,3275,474,63,8,9,0,10028799,4701734,1262208,248624

%N 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.

%H Paul D. Hanna, <a href="/A132623/b132623.txt">Rows n = 0..45, flattened.</a>

%F 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).

%F 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.

%e Triangle begins:

%e 0;

%e 1, 0;

%e 1, 2, 0;

%e 3, 2, 3, 0;

%e 14, 8, 3, 4, 0;

%e 87, 46, 15, 4, 5, 0;

%e 669, 338, 102, 24, 5, 6, 0;

%e 6098, 2992, 861, 188, 35, 6, 7, 0;

%e 64050, 30800, 8589, 1788, 310, 48, 7, 8, 0;

%e 759817, 360110, 98238, 19800, 3275, 474, 63, 8, 9, 0; ...

%e -------------------------------------

%e MATRIX POWER SERIES PROPERTY.

%e [I - T]^-1 = Sum_{n>=0} T^n and equals T shifted up 1 row

%e (with '1's in the main diagonal):

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 14, 8, 3, 1;

%e 87, 46, 15, 4, 1;

%e 669, 338, 102, 24, 5, 1; ...

%e -------------------------------------

%e GENERATE T FROM MATRIX POWERS OF T.

%e Matrix square T^2 begins:

%e 0;

%e 0, 0;

%e 2, 0, 0;

%e 5, 6, 0, 0;

%e 23, 14, 12, 0, 0;

%e 143, 78, 27, 20, 0, 0; ...

%e so that T(4,1) = T(3,1) + [T^2](3,1) = 2 + 6 = 8;

%e and T(3,0) = T(2,0) + [T^2](2,0) = 1 + 2 = 3.

%e Matrix cube T^3 begins:

%e 0;

%e 0, 0;

%e 0, 0, 0;

%e 6, 0, 0, 0;

%e 26, 24, 0, 0, 0;

%e 165, 94, 60, 0, 0, 0; ...

%e so that T(5,1) = T(4,1) + [T^2](4,1) + [T^3](4,1) = 8 + 14 + 24 = 46;

%e and T(4,0) = T(3,0) + [T^2](3,0) + [T^3](3,0) = 3 + 5 + 6 = 14.

%e -------------------------------------

%e ILLUSTRATE G.F. FOR COLUMN k:

%e 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)) +...

%e 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)) +...

%e 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)) +...

%o (PARI) /* Using the matrix power formula: */

%o 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]) )))

%o (PARI) /* Using the g.f. formula for columns: */

%o 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))

%o for(n=0,15, for(k=0,n, print1(T(n,k),", ")); print(""))

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

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Aug 25 2007