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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-Fran├žois Alcover, Dec 06 2012, translated from Pari *)

PROG

(PARI) T(n, k)=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)+2*sum(j=k-1, 2*n-1-k, T(n-1, j)), T(n, 2*n-k)))))

(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

CROSSREFS

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

Cf. A210111 (left half).

Sequence in context: A091623 A215474 A059616 * A291665 A181836 A124740

Adjacent sequences:  A125050 A125051 A125052 * A125054 A125055 A125056

KEYWORD

nonn,tabf,nice

AUTHOR

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

STATUS

approved

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Last modified October 21 17:10 EDT 2018. Contains 316427 sequences. (Running on oeis4.)