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A133815 Square array of Hankel transforms of binomial(n+k,floor((n+k)/2)), read by antidiagonals. 0
1, 1, 1, 1, 1, 1, 1, -1, 2, 1, 1, -1, 3, 3, 1, 1, 1, 4, -6, 6, 1, 1, 1, 5, -10, 20, 10, 1, 1, -1, 6, 15, 50, -50, 20, 1, 1, -1, 7, 21, 105, -175, 175, 35, 1, 1, 1, 8, -28, 196, 490, 980, -490, 70, 1, 1, 1, 9, -36, 336, 1176, 4116, -4116, 1764, 126, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

T(n+1,k) is the Hankel transform of binomial(n+k,floor((n+k)/2)).

Even-indexed columns count tilings of hexagons: A002415 (<2,n,2>), A047819 (<3,n,3>), A047835 (<4,n,4>), etc.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(n,k) = if(k mod 2 = 0, Product_{j=0..(k-2)/2} C(n+k/2+j,k/2)/C(k/2+j,k/2),(cos(Pi*n/2)+sin(Pi*n/2))*Product_{j=0..(k-3)/2} C(n+(k+1)/2+j,(k+1)/2)/C((k+1)/2+j,(k+1)/2)}).

EXAMPLE

Array begins

  1,    1,    1,    1,    1,    1, ...

  1,    1,    2,    3,    6,   10, ...

  1,   -1,    3,   -6,   20,  -50, ...

  1,   -1,    4,  -10,   50, -175, ...

  1,    1,    5,   15,  105,  490, ...

  1,    1,    6,   21,  196, 1176, ...

As a number triangle, T(n-k,k) gives

  1;

  1,   1;

  1,   1,   1;

  1,  -1,   2,   1;

  1,  -1,   3,   3,   1;

  1,   1,   4,  -6,   6,   1;

  1,   1,   5, -10,  20,  10,   1;

  1,  -1,   6,  15,  50, -50,  20,   1;

MATHEMATICA

T[ n_, m_] := With[{k = Quotient[m + 1, 2]}, (-1)^(Quotient[n, 2] m) Product[ Binomial[n + k + j, k] / Binomial[k + j, k], {j, 0, k - 1 - Mod[m, 2]}]];

(* Michael Somos, Apr 03 2021 *)

PROG

(PARI) alias(C, binomial);

T(n, k) = if (k % 2 == 0, prod(j=0, (k-2)/2, C(n+k/2+j, k/2)/C(k/2+j, k/2)), (cos(Pi*n/2)+sin(Pi*n/2))*prod(j=0, (k-3)/2, C(n+(k+1)/2+j, (k+1)/2)/C((k+1)/2+j, (k+1)/2)));

tabl(nn) = matrix(nn, nn, n, k, round(T(n-1, k-1))); \\ Michel Marcus, Dec 10 2016

(PARI) T(n, m) = my(k = (m+1)\2); (-1)^(n\2*m) * prod(j=0, k-1-m%2, binomial(n+k+j, k) / binomial(k+j, k)); /* Michael Somos, Apr 03 2021 */

CROSSREFS

Cf. A103905, A120247.

Sequence in context: A293551 A099233 A303912 * A305027 A335570 A323718

Adjacent sequences:  A133812 A133813 A133814 * A133816 A133817 A133818

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, Sep 24 2007

STATUS

approved

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Last modified December 2 16:17 EST 2021. Contains 349445 sequences. (Running on oeis4.)