A126150 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301.

1, 1, 4, 1, 6, 24, 36, 24, 6, 96, 384, 636, 744, 636, 384, 96, 2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, 2976, 151416, 605664, 1042056, 1407024, 1650456, 1736064, 1650456, 1407024, 1042056, 605664, 151416, 11449296, 45797184

Offset

0,3

Links

Table of n, a(n) for n=0..37.

Formula

Sum_{k=0,2n} (-1)^k*C(2n,k)*T(n,k) = (-6)^n.

Example

Triangle begins:
1;
1, 4, 1;
6, 24, 36, 24, 6;
96, 384, 636, 744, 636, 384, 96;
2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, 2976;
151416, 605664, 1042056, 1407024, 1650456, 1736064, 1650456, 1407024, 1042056, 605664, 151416; ...
If we write the triangle like this:
.......................... ....1;
................... ....1, ....4, ....1;
............ ....6, ...24, ...36, ...24, ....6;
..... ...96, ..384, ..636, ..744, ..636, ..384, ...96;
.2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, .2976;
then the first term in each row is the sum of the previous row:
2976 = 96 + 384 + 636 + 744 + 636 + 384 + 96
the next term is 4 times the first:
11904 = 4*2976,
and the remaining terms in each row are obtained by the rule
illustrated by:
20256 = 2*11904 - 2976 - 6*96;
26304 = 2*20256 - 11904 - 6*384;
28536 = 2*26304 - 20256 - 6*636;
26304 = 2*28536 - 26304 - 6*744;
20256 = 2*26304 - 28536 - 6*636;
11904 = 2*20256 - 26304 - 6*384;
2976 = 2*11904 - 20256 - 6*96.
An alternate recurrence is illustrated by:
11904 = 2976 + 3*(96 + 384 + 636 + 744 + 636 + 384 + 96);
20256 = 11904 + 3*(384 + 636 + 744 + 636 + 384);
26304 = 20256 + 3*(636 + 744 + 636);
28536 = 26304 + 3*(744);
and then for k>n, T(n,k) = T(n,2n-k).

Prog

(PARI) T(n, k)=local(p=3); 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=3); 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)))))

Crossrefs

Cf. A126151 (column 0); diagonals: A126152, A126153; A126154; variants: A008301, A125053, A126155.

Sequence in context: A343599 A191714 A370356 * A364509 A349545 A291056

Adjacent sequences: A126147 A126148 A126149 * A126151 A126152 A126153

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 19 2006

Status

approved