A126155 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301. Second term 5 times first term.

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

Offset: 0

Author

Paul D. Hanna, Dec 20 2006

Keywords

• nonn
• tabl

Examples

			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).
		

Crossrefs

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

Programs

Maple

T := proc(n,k) option remember; local j;
  if n = 1 then 1
elif k = 1 then add(T(n-1, j), j=1..2*n-3)
elif k = 2 then 5*T(n, 1)
elif k > n then T(n, 2*n-k)
else 2*T(n, k-1)-T(n, k-2)-8*T(n-1, k-2)
  fi end:
seq(print(seq(T(n,k), k=1..2*n-1)), n=1..6); # Peter Luschny, May 12 2014

Mathematica

T[n_, k_] := T[n, k] = Which[n==1, 1, k==1, Sum[T[n-1, j], {j, 1, 2n-3}], k==2, 5 T[n, 1], k>n, T[n, 2n-k], True, 2 T[n, k-1] - T[n, k-2] - 8 T[n-1, k-2]];
Table[T[n, k], {n, 1, 6}, {k, 1, 2n-1}] (* Jean-François Alcover, Jun 15 2019, from Maple *)

PARI

{T(n,k) = local(p=4);if(2*n
				

PARI

/* Alternate Recurrence: */
{T(n,k) = local(p=4);if(2*n
				

SageMath

from functools import cache
@cache
def R(n, k):
      return (1 if n == 1 else sum(R(n-1, j) for j in range(1, 2*n-2))
                if k == 1 else 5*R(n, 1) if k == 2 else R(n, 2*n-k)
                if k > n else 2*R(n, k-1) - R(n, k-2) - 8*R(n-1, k-2))
def A126155(n, k): return R(n+1, k+1)
for n in range(5): print([A126155(n, k) for k in range(2*n+1)])
# Peter Luschny, Dec 14 2023

Formula

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