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A053445
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Second differences of partition numbers A000041.
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30
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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
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, 3.
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LINKS
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FORMULA
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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
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EXAMPLE
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a(8) = 7 - 4 = 3; the corresponding partitions are 44, 332 and 2222.
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MAPLE
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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):
# 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
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MATHEMATICA
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Differences[PartitionsP[Range[0, 60]], 2] (* Harvey P. Dale, Jan 29 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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EXTENSIONS
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Start of sequence changed, Apr 25 2003
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STATUS
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approved
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