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A126155 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..35.

FORMULA

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

EXAMPLE

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).

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

PROG

(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))))))}

for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))

(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)))))}

for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))

CROSSREFS

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

Sequence in context: A193860 A211849 A222182 * A021197 A073116 A201525

Adjacent sequences:  A126152 A126153 A126154 * A126156 A126157 A126158

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 20 2006

STATUS

approved

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Last modified December 2 23:29 EST 2016. Contains 278694 sequences.