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A000990
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Number of plane partitions of n with at most two rows.
(Formerly M2462 N0978)
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19
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1, 1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, 605, 895, 1291, 1866, 2648, 3760, 5260, 7352, 10160, 14008, 19140, 26085, 35277, 47575, 63753, 85175, 113175, 149938, 197686, 259891, 340225, 444135, 577593, 749131, 968281, 1248320, 1604340, 2056809, 2629357, 3353404
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OFFSET
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0,3
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COMMENTS
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Also number of partitions of n into parts of 2 kinds except for 1. - Reinhard Zumkeller, Nov 06 2012
Antidiagonal sums of triangle A093010.
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REFERENCES
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G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 105.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.7).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: 1 / ( (1-x) * Product_{m>=2} (1-x^m)^2 ) = (1-x) / Product_{m>=1} (1-x^m)^2.
G.f.: exp( Sum_{n>=1} ((1+x^n)/(1-x^n))*x^n/n ). - Paul D. Hanna, Apr 22 2010
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Oct 28 2015
G.f.: exp(Sum_{k>=1} (2*sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(min(i, 2)+j-1, j)*
b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[Min[i, 2]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
Flatten[{1, Differences[Table[Sum[PartitionsP[j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 50}]]}] (* Vaclav Kotesovec, Oct 28 2015 *)
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff((1-x)/prod(k=1, n, 1-x^k, 1+x*O(x^n))^2, n)) /* Michael Somos, Jan 29 2005 */
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, ((1+x^m)/(1-x^m+x*O(x^n)))*x^m/m)), n)} \\ Paul D. Hanna, Apr 22 2010
(Haskell)
a000990 = p $ tail a008619_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(PARI) x='x+O('x^66); Vec((1-x)/eta(x)^2) \\ Joerg Arndt, May 01 2013
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(&*[1-x^j: j in [1..2*m]] )^2 )); // G. C. Greubel, Dec 06 2018
(Sage) s=((1-x)/prod(1-x^j for j in (1..60))^2).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 06 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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