A125053 Variant of triangle A008301, read by rows of 2*n+1 terms, such that the first column is the secant numbers (A000364).

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

Offset: 0

Author

Paul D. Hanna, Nov 21 2006, Dec 20 2006

Keywords

• nonn
• tabf
• nice

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).

Examples

			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).
		

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;
• Foata, Dominique; Han, Guo-Niu; Strehl, Volker The Entringer-Poupard matrix sequence. Linear Algebra Appl. 512, 71-96 (2017).

Crossrefs

Cf. A008301, A000364 (secant numbers, which are the row sums), A125054 (central terms), A125055, A000182, A008282.

Cf. A210111 (left half).

Programs

Haskell

a125053 n k = a125053_tabf !! n !! k
a125053_row n = a125053_tabf !! n
a125053_tabf = iterate f [1] where
f zs = zs' ++ reverse (init zs') where
zs' = (sum zs) : g (map (* 2) zs) (sum zs)
g [x] y = [x + y]
g xs y = y' : g (tail $ init xs) y' where y' = sum xs + y
-- Reinhard Zumkeller, Mar 17 2012

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 *)

PARI

T(n,k)=if(2*n
				

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.