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%I #75 Aug 27 2023 19:43:58
%S 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,
%T 139,192,199,269,287,373,406,521,566,718,792,983,1092,1346,1496,1827,
%U 2045,2465,2772,3323,3733,4449,5016,5929,6696,7882,8897,10426
%N Second differences of partition numbers A000041.
%C First differences of 0 1 1 2 2 4 4 7 8 12 14 21 24 34 41 55 ... (A002865).
%C 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; ...
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, 3.
%H Andrew van den Hoeven, <a href="/A053445/b053445.txt">Table of n, a(n) for n = 0..9998</a> (terms up to a(1000) from T. D. Noe)
%H Gert Almkvist, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6126.pdf">On the differences of the partition function</a>, Acta Arith., 61.2 (1992), 173-181.
%H K. Banerjee, <a href="https://doi.org/10.54550/ECA2023V3S2R12">Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach</a>, Enum. Combin. Applic. 3 (2023) # S2R12
%H P. Furlan, G. M. Sotkov and I. T. Todorov, <a href="http://dx.doi.org/10.1007/BF02742979">Two-Dimensional Conformal Quantum Field Theory</a>, Rivista d. Nuovo Cimento 12, 6 (1989) 1-202.
%H Cristiano Husu, <a href="https://arxiv.org/abs/1804.09883">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</a>, arXiv:1804.09883 [math.NT], 2018.
%F a(n) = A002865(n+2) - A002865(n+1).
%F 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
%F G.f.: 1/x - 1/x^2 + ((1 - x)/x^2)*Product_{k>=2} 1/(1 - x^k). - _Ilya Gutkovskiy_, Feb 28 2017
%e a(8) = 7 - 4 = 3; the corresponding partitions are 44, 332 and 2222.
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p end:
%p a:= n-> b(n$2) -2*b(n+1$2) +b(n+2$2):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, May 19 2014
%p # alternative Maple program:
%p P:= [seq(combinat:-numbpart(n),n=0..1002)]:
%p seq(P[i]-2*P[i+1]+P[i+2],i=1..1001); # _Robert Israel_, Dec 15 2014
%t Table[(PartitionsP[n+2]-PartitionsP[n+1])-(PartitionsP[n+1]-PartitionsP[n]),{n,0,42}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t Differences[Table[ PartitionsP[n], {n, 0, 56}], 2] (* _Jean-François Alcover_, Sep 07 2011 *)
%t Differences[PartitionsP[Range[0,60]],2] (* _Harvey P. Dale_, Jan 29 2016 *)
%o (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
%o (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
%o (Python)
%o from sympy import npartitions
%o def A053445(n): return npartitions(n+2)-(npartitions(n+1)<<1)+npartitions(n) # _Chai Wah Wu_, Mar 30 2023
%Y Cf. A000041, A002865, A072380, A081094, A081095.
%K easy,nice,nonn
%O 0,5
%A _Alford Arnold_, Jan 12 2000
%E More terms from _James A. Sellers_, Feb 02 2000
%E Start of sequence changed, Apr 25 2003
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