A126150 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301. Second term 4 times first term.

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

Author

Paul D. Hanna, Dec 19 2006

Keywords

• nonn
• tabl

Examples

			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).
		

Crossrefs

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

Programs

PARI

T(n,k)=local(p=3);if(2*n
				

PARI

/* Alternate Recurrence: */ T(n,k)=local(p=3);if(2*n
				

Formula

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