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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 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

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