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

0,4

Links

Table of n, a(n) for n=0..49.

Formula

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

Example

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.

Prog

(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(""))

Crossrefs

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

Sequence in context: A158835 A236961 A245958 * A136212 A136215 A136737

Adjacent sequences: A138268 A138269 A138270 * A138272 A138273 A138274

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 11 2008

Status

approved