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A003116
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Expansion of the reciprocal of the g.f. defining A039924.
(Formerly M1068)
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24
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1, 1, 2, 4, 7, 13, 23, 41, 72, 127, 222, 388, 677, 1179, 2052, 3569, 6203, 10778, 18722, 32513, 56455, 98017, 170161, 295389, 512755, 890043, 1544907, 2681554, 4654417, 8078679, 14022089, 24337897, 42242732, 73319574, 127258596, 220878683
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OFFSET
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0,3
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COMMENTS
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Conjecture: a(n) is the number of compositions p(1) + p(2) + ... + p(m) = n with p(i)-p(i-1) <= 1, see example; cf. A034297. - Vladeta Jovovic, Feb 09 2004
Former definition was "Expansion of reciprocal of a determinant." - N. J. A. Sloane, Aug 10 2018
Jovovic's conjecture can be proved as follows. There is a sign-changing involution defined on pairs (L1,L2) where L1 is a partition with difference >= 2 between consecutive parts and L2 is the number of compositions described by Jovovic, with the sign (-1)^(Number of parts of L1).
Let a be the largest part of L1 and b the largest part of L2. If b-a>=2 then move b from L2 to the top of L1, otherwise move a to the top of L2.
Since this is an involution and it changes the sign (the number of parts of L1 changes parity) this proves it, since the g.f. of A039924 is exactly the signed-enumeration of the set given by L1. (End)
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REFERENCES
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D. H. Lehmer, Combinatorial and cyclotomic properties of certain tridiagonal matrices. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 53-74. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. MR0441852.
H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
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/(Sum_{k>=0} x^(k^2)(-1)^k/(Product_{i=1..k} 1-x^i)).
a(n) ~ c * d^n, where d = 1/A347901 = 1.73566282453034742565826074971966853... and c = 0.9180565304926754125870866477349969555868577236908640010903420353... - Vaclav Kotesovec, Nov 01 2021
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EXAMPLE
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There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)-p(k-1) <= 1:
[ 1] [ 1 1 1 1 1 1 ]
[ 2] [ 1 1 1 1 2 ]
[ 3] [ 1 1 1 2 1 ]
[ 4] [ 1 1 2 1 1 ]
[ 5] [ 1 1 2 2 ]
[ 6] [ 1 2 1 1 1 ]
[ 7] [ 1 2 1 2 ]
[ 8] [ 1 2 2 1 ]
[ 9] [ 1 2 3 ]
[10] [ 2 1 1 1 1 ]
[11] [ 2 1 1 2 ]
[12] [ 2 1 2 1 ]
[13] [ 2 2 1 1 ]
[14] [ 2 2 2 ]
[15] [ 2 3 1 ]
[16] [ 3 1 1 1 ]
[17] [ 3 1 2 ]
[18] [ 3 2 1 ]
[19] [ 3 3 ]
[20] [ 4 1 1 ]
[21] [ 4 2 ]
[22] [ 5 1 ]
[23] [ 6 ]
Replacing the condition with p(k)-p(k-1) <= 0 gives integer partitions.
(End)
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MATHEMATICA
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max = 35; f[x_] := 1/Sum[x^k^2*((-1)^k/Product[1 - x^i, {i, 1, k}]), {k, 0, Floor[Sqrt[max]]}]; CoefficientList[ Series[f[x], {x, 0, max}], x](* Jean-François Alcover, Jun 12 2012, after PARI *)
b[n_, k_] := b[n, k] = Expand[If[n == 0, 1, x*
Sum[b[n - j, j], {j, 1, Min[n, k + 1]}]]];
a[n_] := Total@CoefficientList[b[n, n], x];
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff(1/sum(k=0, sqrtint(n), x^k^2/prod(i=1, k, x^i-1, 1+x*O(x^n))), n))
(Haskell)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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