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A003293
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Number of planar partitions of n decreasing across rows.
(Formerly M1058)
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2
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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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).
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LINKS
| Nathaniel Johnston, Table of n, a(n) for n = 0..1000
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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
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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]
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MATHEMATICA
| CoefficientList[Series[Product[(1-x^k)^(-Ceiling[k/2]),
{k, 1, 40}], {x, 0, 40}], x][[1 ;; 40]]
(* From Jean-François Alcover, Apr 18 2011, after M. Somos *)
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PROG
| (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 */
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CROSSREFS
| Sequence in context: A168368 A182746 A100482 * A192759 A094974 A189593
Adjacent sequences: A003290 A003291 A003292 * A003294 A003295 A003296
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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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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