0,5

First differences of 0 1 1 2 2 4 4 7 8 12 14 21 24 34 41 55 ... (A002865).

For n > 2, a(n-2) is the number of partitions of n with all parts > 1 and with the largest part occurring more than once. The list of partitions counted begins 22 (so a(2) = 1); 33, 222 (so a(4) = 2); 44, 332, 2222 (so a(6) = 3); 333; 55, 442, 3322, 22222; 443, 3332; 66, 552, 444, 4422, 3333, 33222, 222222; 553, 4432, 33322; ...

a(n) is the number of certain level-n quasi-primary states of a quotient space of certain Verma modules. See the Furlan et al. reference p. 67. - Wolfdieter Lang, Apr 25 2003

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, 3.

Andrew van den Hoeven, Table of n, a(n) for n = 0..9998 (terms up to a(1000) from T. D. Noe)

Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.

K. Banerjee, Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach, Enum. Combin. Applic. 3 (2023) # S2R12

P. Furlan, G. M. Sotkov and I. T. Todorov, Two-Dimensional Conformal Quantum Field Theory, Rivista d. Nuovo Cimento 12, 6 (1989) 1-202.

Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.

a(n) = A002865(n+2) - A002865(n+1).

a(n) ~ exp(sqrt(2*n/3)*Pi)*Pi^2/(24*sqrt(3)*n^2) * (1 + (23*Pi/(24*sqrt(6)) - 3*sqrt(6)/Pi)/sqrt(n) + (625*Pi^2/6912 + 45/(2*Pi^2) - 115/24)/n). - Vaclav Kotesovec, Nov 04 2016

G.f.: 1/x - 1/x^2 + ((1 - x)/x^2)*Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Feb 28 2017

a(8) = 7 - 4 = 3; the corresponding partitions are 44, 332 and 2222.

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

b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

end:

a:= n-> b(n$2) -2*b(n+1$2) +b(n+2$2):

seq(a(n), n=0..80); # Alois P. Heinz, May 19 2014

# Alternative:

P:= [seq(combinat:-numbpart(n), n=0..1002)]:

seq(P[i]-2*P[i+1]+P[i+2], i=1..1001); # Robert Israel, Dec 15 2014

Table[(PartitionsP[n+2]-PartitionsP[n+1])-(PartitionsP[n+1]-PartitionsP[n]), {n, 0, 42}] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)

Differences[Table[ PartitionsP[n], {n, 0, 56}], 2] (* Jean-François Alcover, Sep 07 2011 *)

Differences[PartitionsP[Range[0, 60]], 2] (* Harvey P. Dale, Jan 29 2016 *)

(Magma) m:=58; S:=[ NumberOfPartitions(n): n in [0..m] ]; [ S[n+2]-2*S[n+1]+S[n]: n in [1..m-2] ]; // Klaus Brockhaus, Jun 09 2009

(PARI) lista(nn) = {v = vector(nn, n, numbpart(n-1)); dv = vector(#v-1, n, v[n+1] - v[n]); vector(#dv-1, n, dv[n+1] - dv[n]); } \\ Michel Marcus, Dec 15 2014

(Python)

from sympy import npartitions

def A053445(n): return npartitions(n+2)-(npartitions(n+1)<<1)+npartitions(n) # Chai Wah Wu, Mar 30 2023

Cf. A000041, A002865, A072380, A081094, A081095.

Sequence in context: A263352 A008734 A226649 * A175990 A290982 A346034

Adjacent sequences: A053442 A053443 A053444 * A053446 A053447 A053448

easy,nice,nonn

Alford Arnold, Jan 12 2000

More terms from James Sellers, Feb 02 2000

Start of sequence changed, Apr 25 2003

approved