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A003293 Number of planar partitions of n decreasing across rows.
(Formerly M1058)
+0
2
1, 1, 2, 4, 7, 12, 21, 34, 56, 90, 143, 223, 348, 532, 811, 1224, 1834, 2725, 4031, 5914, 8638, 12540, 18116, 26035, 37262, 53070, 75292, 106377, 149738, 209980, 293473, 408734, 567484, 785409, 1083817, 1491247, 2046233, 2800125, 3821959, 5203515 (list; graph; listen)
OFFSET

0,3

COMMENT

Also number of planar partitions monotonically decreasing down anti-diagonals (i.e., with b(n,k)<=b(n-1,k+1)). Transpose (to get planar partitions decreasing down columns), then take the conjugate of each row. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 15 2006

REFERENCES

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 133.

M. S. Cheema and W. E. Conway, Numerical investigation of certain asymptotic results in the theory of partitions, Math. Comp., 26 (1972), 999-1005.

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

FORMULA

G.f.: Product (1 - x^k )^{-c(k)}, c(k) = 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ....

Euler transform of A110654. - Michael Somos Sep 19 2006

MAPLE

with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr (n-> `if`(modp(n, 2)=0, n, n+1)/2): seq (a(n), n=0..39); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 08 2008]

PROGRAM

(PARI) {a(n)=if(n<0, 0, polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-ceil(k/2)), n))} /* Michael Somos Sep 19 2006 */

CROSSREFS

Sequence in context: A079970 A079816 A100482 this_sequence A094974 A100671 A005251

Adjacent sequences: A003290 A003291 A003292 this_sequence A003294 A003295 A003296

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

EXTENSIONS

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 06 2000. Additional comments from Michael Somos, May 19, 2000.

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Last modified December 20 00:58 EST 2009. Contains 171054 sequences.


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