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A116931 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least twice. 5
1, 2, 2, 4, 4, 8, 8, 13, 15, 22, 24, 37, 40, 57, 64, 89, 98, 135, 149, 199, 224, 292, 325, 424, 472, 601, 676, 850, 950, 1191, 1329, 1643, 1845, 2258, 2524, 3082, 3442, 4158, 4659, 5591, 6246, 7472, 8338, 9903, 11072, 13077, 14586, 17184, 19150, 22431, 25019 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also, partitions of n in which any two distinct parts differ by at least 2. Example: a(5) = 4 because we have [5], [4,1], [3,1,1] and [1,1,1,1,1].

REFERENCES

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 52, Article 298.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..7500

FORMULA

G.f.: sum(x^k*product(1+x^(2j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b.

EXAMPLE

a(5) = 4 because we have [5], [3,1,1], [2,1,1,1] and [1,1,1,1,1].

q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 13*q^8 + 15*q^9 + ...

There are a(9) = 15 partitions of 9 where distinct parts differ by at least 2:

01:  [ 1 1 1 1 1 1 1 1 1 ]

02:  [ 3 1 1 1 1 1 1 ]

03:  [ 3 3 1 1 1 ]

04:  [ 3 3 3 ]

05:  [ 4 1 1 1 1 1 ]

06:  [ 4 4 1 ]

07:  [ 5 1 1 1 1 ]

08:  [ 5 2 2 ]

09:  [ 5 3 1 ]

10:  [ 6 1 1 1 ]

11:  [ 6 3 ]

12:  [ 7 1 1 ]

13:  [ 7 2 ]

14:  [ 8 1 ]

15:  [ 9 ]

- Joerg Arndt, Jun 09 2013

MAPLE

g:=sum(x^k*product(1+x^(2*j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..70): gser:=series(g, x=0, 60): seq(coeff(gser, x^n), n=1..54);

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

       b(n, i-1) +add(b(n-i*j, i-2), j=1..n/i)))

    end:

a:= n-> b(n, n):

seq(a(n), n=1..70);  # Alois P. Heinz, Nov 04 2012

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-2], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-Fran├žois Alcover, Mar 24 2015, after Alois P. Heinz *)

PROG

(PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k) * prod( j=1, k-1, 1 + x^(2*j) / (1 - x^j), 1 + x * O(x^(n-k)))), n))} /* Michael Somos, Jan 26 2008 */

CROSSREFS

Cf. A116932.

Column k=2 of A218698. - Alois P. Heinz, Nov 04 2012

Column k=0 of A268193. - Alois P. Heinz, Feb 13 2016

Sequence in context: A046971 A051754 A108747 * A206558 A145810 A172148

Adjacent sequences:  A116928 A116929 A116930 * A116932 A116933 A116934

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Feb 27 2006

STATUS

approved

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Last modified March 23 14:17 EDT 2019. Contains 321431 sequences. (Running on oeis4.)