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 A125053 Variant of triangle A008301, read by rows of 2*n+1 terms, such that the first column is the secant numbers (A000364). 8
 1, 1, 3, 1, 5, 15, 21, 15, 5, 61, 183, 285, 327, 285, 183, 61, 1385, 4155, 6681, 8475, 9129, 8475, 6681, 4155, 1385, 50521, 151563, 247065, 325947, 378105, 396363, 378105, 325947, 247065, 151563, 50521, 2702765, 8108295, 13311741, 17908935 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Foata and Han refer to this as the triangle of Poupard numbers h_n(k). - N. J. A. Sloane, Feb 17 2014 Central terms (A125054) equal the binomial transform of the tangent numbers (A000182). LINKS Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, Nov 20 2013; FORMULA Sum_{k=0..2n} C(2n,k)*T(n,k) = 4^n * A000182(n), where A000182 are the tangent numbers. Sum_{k=0..2n} (-1)^n*C(2n,k)*T(n,k) = (-4)^n. EXAMPLE If we write the triangle like this: ......................... ...1; ................... ...1, ...3, ...1; ............. ...5, ..15, ..21, ..15, ...5; ....... ..61, .183, .285, .327, .285, .183, ..61; . 1385, 4155, 6681, 8475, 9129, 8475, 6681, 4155, 1385; then the first nonzero term is the sum of the previous row: 1385 = 61 + 183 + 285 + 327 + 285 + 183 + 61, the next term is 3 times the first: 4155 = 3*1385, and the remaining terms in each row are obtained by the rule illustrated by: 6681 = 2*4155 - 1385 - 4*61; 8475 = 2*6681 - 4155 - 4*183; 9129 = 2*8475 - 6681 - 4*285; 8475 = 2*9129 - 8475 - 4*327; 6681 = 2*8475 - 9129 - 4*285; 4155 = 2*6681 - 8475 - 4*183; 1385 = 2*4155 - 6681 - 4*61. An alternate recurrence is illustrated by: 4155 = 1385 + 2*(61 + 183 + 285 + 327 + 285 + 183 + 61); 6681 = 4155 + 2*(183 + 285 + 327 + 285 + 183); 8475 = 6681 + 2*(285 + 327 + 285); 9129 = 8475 + 2*(327); and then for k>n, T(n,k) = T(n,2*n-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 3*T(n, 1) elif k > n then T(n, 2*n-k) else 2*T(n, k-1) - T(n, k-2) - 4*T(n-1, k-2)   fi end: seq(print(seq(T(n, k), k=1..2*n-1)), n=1..5); # Peter Luschny, May 11 2014 MATHEMATICA t[n_, k_] := t[n, k] = If[2*n < k || k < 0, 0, If[n == 0 && k == 0, 1, If[k == 0, Sum[t[n-1, j], {j, 0, 2*n-2}], If[k <= n, t[n, k-1] + 2*Sum[t[n-1, j], {j, k-1, 2*n-1-k}], t[n, 2*n-k]]]]]; Table[t[n, k], {n, 0, 6}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Dec 06 2012, translated from Pari *) PROG (PARI) T(n, k)=if(2*n

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Last modified January 17 14:12 EST 2019. Contains 319225 sequences. (Running on oeis4.)