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A005158 Number of alternating sign n X n matrices invariant under a half-turn.
(Formerly M0902)
4
1, 2, 3, 10, 25, 140, 588, 5544, 39204, 622908, 7422987, 198846076 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

LINKS

Table of n, a(n) for n=1..12.

G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184

D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math.CO/0008045

R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

FORMULA

Robbins gives a simple (conjectured) formula.

MATHEMATICA

a[n_?OddQ] := a[n] = (a[n-1]*Binomial[(3*(n-1))/2, (n-1)/2])/ Binomial[n-1, (n-1)/2]; a[n_?EvenQ] := a[n] = (4*a[n-1]*Binomial[(3*n)/2, n/2])/ (3*Binomial[n, n/2]); a[1] = 1; Table[a[n], {n, 1, 11}] (* Jean-Fran├žois Alcover, Sep 12 2012, from Robbins's conjectured(!) formula *)

PROG

(PARI) /* from Robbins's conjectured(!) formula */

/* using memoization for efficiency */

N=33;  v=vector(N); v[1]=1;

{ for (n=2, N,

    if (bitand(n, 1),

        v[n] = (v[n-1]*binomial((3*(n-1))/2, (n-1)/2)) / binomial(n-1, (n-1)/2);

    ,

        v[n] = (4*v[n-1]*binomial((3*n)/2, n/2))/ (3*binomial(n, n/2));

    );

); }

v /* show terms */

/* Joerg Arndt, Sep 12 2012 */

CROSSREFS

a(2n) gives A059475.

Sequence in context: A123029 A103018 A246437 * A182926 A005225 A211208

Adjacent sequences:  A005155 A005156 A005157 * A005159 A005160 A005161

KEYWORD

nonn,nice,more

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 15 04:00 EST 2018. Contains 317225 sequences. (Running on oeis4.)