A053445 Second differences of partition numbers A000041.

1, 0, 1, 0, 2, 0, 3, 1, 4, 2, 7, 3, 10, 7, 14, 11, 22, 17, 32, 28, 45, 43, 67, 63, 95, 96, 134, 139, 192, 199, 269, 287, 373, 406, 521, 566, 718, 792, 983, 1092, 1346, 1496, 1827, 2045, 2465, 2772, 3323, 3733, 4449, 5016, 5929, 6696, 7882, 8897, 10426

Offset: 0

Author

Alford Arnold, Jan 12 2000

Keywords

• easy
• nice
• nonn

Comments

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

Examples

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

References

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

Links

• 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.

Crossrefs

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

Programs

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

Maple

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 Maple program:
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

Mathematica

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 *)

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

Formula

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

Extensions

More terms from James Sellers, Feb 02 2000

Start of sequence changed, Apr 25 2003