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A243920 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) = 2*n+1 and T(n,n)=0 for n>=0, where T^j denotes the j-th matrix power of T. 3
0, 1, 0, 1, 3, 0, 4, 3, 5, 0, 27, 18, 5, 7, 0, 254, 159, 40, 7, 9, 0, 3048, 1836, 435, 70, 9, 11, 0, 44328, 26028, 5930, 903, 108, 11, 13, 0, 755681, 435804, 96640, 14168, 1611, 154, 13, 15, 0, 14750646, 8395065, 1825600, 260484, 28566, 2607, 208, 15, 17, 0, 323999500 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
G.f. of column k: (2*k+1)*x^(k+1) = Sum_{n>=0} T(n,k) * x^n * (1-x)^(n-k) / Product_{j=k+1..n-1} (1+2*j*x).
T(n,k) = [x^n] { (2*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+2*j*x) } for n>k with T(n,k)=0 when k>=n.
EXAMPLE
Triangle begins:
0;
1, 0;
1, 3, 0;
4, 3, 5, 0;
27, 18, 5, 7, 0;
254, 159, 40, 7, 9, 0;
3048, 1836, 435, 70, 9, 11, 0;
44328, 26028, 5930, 903, 108, 11, 13, 0;
755681, 435804, 96640, 14168, 1611, 154, 13, 15, 0;
14750646, 8395065, 1825600, 260484, 28566, 2607, 208, 15, 17, 0;
323999500, 182556012, 39122945, 5471508, 584280, 51524, 3939, 270, 17, 19, 0; ...
-------------------------------------
MATRIX POWER SERIES PROPERTY.
Let T equal this triangle as an infinite triangular matrix; then
[I - T]^(-1) = Sum_{n>=0} T^n and equals T shifted up 1 row
(with all '1's replacing the main diagonal):
1;
1, 1;
4, 3, 1;
27, 18, 5, 1;
254, 159, 40, 7, 1;
3048, 1836, 435, 70, 9, 1;
44328, 26028, 5930, 903, 108, 11, 1; ...
-------------------------------------
GENERATE T FROM MATRIX POWERS OF T.
Matrix square T^2 begins:
0;
0, 0;
3, 0, 0;
8, 15, 0, 0;
51, 36, 35, 0, 0;
470, 303, 80, 63, 0, 0;
5588, 3426, 835, 140, 99, 0, 0;
80904, 48060, 11150, 1743, 216, 143, 0, 0; ...
so that
T(3,0) = T(2,0) + [T^2](2,0) = 1 + 3 = 4;
T(4,1) = T(3,1) + [T^2](3,1) = 3 + 15 = 18.
Matrix cube T^3 begins:
0;
0, 0;
0, 0, 0;
15, 0, 0, 0;
71, 105, 0, 0, 0;
635, 429, 315, 0, 0, 0;
7494, 4707, 1195, 693, 0, 0, 0;
108336, 65304, 15515, 2513, 1287, 0, 0, 0; ...
so that
T(4,0) = T(3,0) + [T^2](3,0) + [T^3](3,0) = 4 + 8 + 15 = 27;
T(5,1) = T(4,1) + [T^2](4,1) + [T^3](4,1) = 18 + 36 + 105 = 159.
-------------------------------------
ILLUSTRATE G.F. FOR COLUMN k:
k=0: 1*x = T(1,0)*x*(1-x) + T(2,0)*x^2*(1-x)^2/(1+2*1*x) + T(3,0)*x^3*(1-x)^3/((1+2*1*x)*(1+2*2*x)) + T(4,0)*x^4*(1-x)^4/((1+2*1*x)*(1+2*2*x)*(1+2*3*x)) +...
k=1: 3*x^2 = T(2,1)*x^2*(1-x) + T(3,1)*x^3*(1-x)^2/(1+2*2*x) + T(4,1)*x^4*(1-x)^3/((1+2*2*x)*(1+2*3*x)) + T(5,1)*x^5*(1-x)^4/((1+2*2*x)*(1+2*3*x)*(1+2*4*x)) +...
k=2: 5*x^3 = T(3,2)*x^3*(1-x) + T(4,2)*x^4*(1-x)^2/(1+2*3*x) + T(5,2)*x^5*(1-x)^3/((1+2*3*x)*(1+2*4*x)) + T(6,2)*x^6*(1-x)^4/((1+2*3*x)*(1+2*4*x)*(1+2*5*x)) +...
-------------------------------------
PROG
(PARI) {T(n, k)=if(n<k+1, 0, polcoeff((2*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+2*j*x+x*O(x^n))), n))}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
(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, 2*n-1, sum(j=1, n-k-1, (M^j)[n, k+1]) )))}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
Cf. A132623 (variant), A243696 (column 0), A243921, A243922, A243923.
Sequence in context: A104514 A349915 A072480 * A344905 A181839 A100939
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 15 2014
STATUS
approved

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Last modified May 22 12:56 EDT 2024. Contains 372755 sequences. (Running on oeis4.)