0,3

Sum_{k=0,2n} (-1)^k*C(2n,k)*T(n,k) = (-8)^n.

The triangle begins:

1;

1, 5, 1;

7, 35, 55, 35, 7;

139, 695, 1195, 1415, 1195, 695, 139;

5473, 27365, 48145, 63365, 69025, 63365, 48145, 27365, 5473;

357721, 1788605, 3175705, 4343885, 5126905, 5403005, 5126905, 4343885, 3175705, 1788605, 357721; ...

If we write the triangle like this:

.......................... ....1;

................... ....1, ....5, ....1;

............ ....7, ...35, ...55, ...35, ....7;

..... ..139, ..695, .1195, .1415, .1195, ..695, ..139;

.5473, 27365, 48145, 63365, 69025, 63365, 48145, 27365, .5473;

then the first term in each row is the sum of the previous row:

5473 = 139 + 695 + 1195 + 1415 + 1195 + 695 + 139

the next term is 5 times the first:

27365 = 5*5473,

and the remaining terms in each row are obtained by the rule

illustrated by:

48145 = 2*27365 - 5473 - 8*139;

63365 = 2*48145 - 27365 - 8*695;

69025 = 2*63365 - 48145 - 8*1195;

63365 = 2*69025 - 63365 - 8*1415;

48145 = 2*63365 - 69025 - 8*1195;

27365 = 2*48145 - 63365 - 8*695;

5473 = 2*27365 - 48145 - 8*139.

An alternate recurrence is illustrated by:

27365 = 5473 + 4*(139 + 695 + 1195 + 1415 + 1195 + 695 + 139);

48145 = 27365 + 4*(695 + 1195 + 1415 + 1195 + 695);

63365 = 48145 + 4*(1195 + 1415 + 1195);

69025 = 63365 + 4*(1415);

and then for k>n, T(n,k) = T(n,2n-k).

(PARI) {T(n, k)=local(p=4); if(2*n<k|k<0, 0, if(n==0&k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k==1, (p+1)*T(n, 0), if(k<=n, 2*T(n, k-1)-T(n, k-2)-2*p*T(n-1, k-2), T(n, 2*n-k))))))} (PARI) /* Alternate Recurrence: */ {T(n, k)=local(p=4); if(2*n<k|k<0, 0, if(n==0&k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k<=n, T(n, k-1)+p*sum(j=k-1, 2*n-1-k, T(n-1, j)), T(n, 2*n-k)))))}

Cf. A126156 (column 0); diagonals: A126157, A126158; A126159; variants: A008301 (p=1), A125053 (p=2), A126150 (p=3).

Sequence in context: A001945 A051854 A006569 this_sequence A021197 A073116 A115521

Adjacent sequences: A126152 A126153 A126154 this_sequence A126156 A126157 A126158

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 20 2006