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A003116 Expansion of reciprocal of a determinant.
(Formerly M1068)
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

G.f. is reciprocal of g.f. of A039924.

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

Row sums and central terms of the triangle in A168396: a(n) = A168396(2*n+1,n) and for n > 0: a(n) = sum(A168396(n,k): k=1..n). - Reinhard Zumkeller, Sep 13 2013

REFERENCES

D. H. Lehmer, Lecture course on history of mathematics, Univ. Calif. Berkeley, 1973.

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..400

FORMULA

G.f.: 1/(Sum_{k>=0} x^(k^2)(-1)^k/(Product_{i=1..k} 1-x^i)).

EXAMPLE

From Joerg Arndt, Dec 29 2012: (Start)

There are a(6)=23 compositions p(1)+p(2)+...+p(m)=10 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 by p(k)-p(k-1) <= 0 gives integer partitions.

(End)

MATHEMATICA

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-Fran├žois Alcover, Jun 12 2012, after PARI *)

PROG

(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)

a003116 n = a168396 (2 * n + 1) n  -- Reinhard Zumkeller, Sep 13 2013

CROSSREFS

Cf. A003114, A039924.

Cf. A034297.

Sequence in context: A239553 A136299 A208354 * A260917 A165648 A078038

Adjacent sequences:  A003113 A003114 A003115 * A003117 A003118 A003119

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Herman P. Robinson

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.