A138271 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^4 shift right.

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 268, 78, 18, 4, 1, 3164, 798, 156, 28, 5, 1, 43672, 9874, 1714, 268, 40, 6, 1, 682632, 141282, 22368, 3164, 420, 54, 7, 1, 11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1, 224283416, 40400466, 5566728, 682632, 77720

Offset: 0

Author

Paul D. Hanna, Mar 11 2008

Keywords

• nonn
• tabl

Examples

			Triangle T begins:
1;
1, 1;
4, 2, 1;
28, 10, 3, 1;
268, 78, 18, 4, 1;
3164, 798, 156, 28, 5, 1;
43672, 9874, 1714, 268, 40, 6, 1;
682632, 141282, 22368, 3164, 420, 54, 7, 1;
11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1;
224283416, 40400466, 5566728, 682632, 77720, 8378, 868, 88, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^4 shift right:
column 1 of T = column 0 of T^2;
column 2 of T = column 0 of T^3;
column 3 of T = column 0 of T^4.
Matrix square of T, T^2, begins:
1;
2, 1;
10, 4, 1;
78, 26, 6, 1;
798, 232, 48, 8, 1;
9874, 2578, 486, 76, 10, 1;
141282, 33764, 5888, 864, 110, 12, 1;
2273730, 503910, 82210, 11396, 1390, 150, 14, 1; ...
where column k of T^2 = column 1 of T^(k+1):
column 0 of T^2 = column 1 of T;
column 2 of T^2 = column 1 of T^3;
column 3 of T^2 = column 1 of T^4.
Matrix cube of T, T^3, begins:
1;
3, 1;
18, 6, 1;
156, 48, 9, 1;
1714, 486, 90, 12, 1;
22368, 5888, 1050, 144, 15, 1;
333910, 82210, 14046, 1908, 210, 18, 1;
5566728, 1289928, 211182, 28072, 3120, 288, 21, 1; ...
where column k of T^3 = column 2 of T^(k+1):
column 0 of T^3 = column 2 of T;
column 1 of T^3 = column 2 of T^2;
column 3 of T^3 = column 2 of T^4.
Matrix 4th power of T, T^4, begins:
1;
4, 1;
28, 8, 1;
268, 76, 12, 1;
3164, 864, 144, 16, 1;
43672, 11396, 1908, 232, 20, 1;
682632, 170000, 28072, 3520, 340, 24, 1;
11834536, 2814832, 454848, 57408, 5820, 468, 28, 1; ...
where column k of T^4 = column 3 of T^(k+1):
column 0 of T^4 = column 3 of T = column 0 of T shift left;
column 1 of T^4 = column 3 of T^2;
column 2 of T^4 = column 3 of T^3.
		

Crossrefs

Cf. columns: A138272, A138273, A138274; central terms: A138275; variants: A091351, A094587, A135902.

Programs

PARI

{T(n, k) = if(k>n||k<0, 0, if(k==n, 1, if(k==0, T(n+2, 3), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

PARI

/* Column k of T = column 0 of T^(k+1): */
{T(n, k) = local(M=if(n==0,Mat(1),matrix(n,n,r,c,if(r>=c,T(r-1,c-1))))); if(k==n, 1, if(k==0, (M^4)[n, 1],(M^(k+1))[n-k+1, 1]))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.