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 A053445 Second differences of partition numbers A000041. 29
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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 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. P. Furlan, G. M. Sotkov and I. T. Todorov, Two-Dimensional Conformal Quantum Field Theory, Rivista d. Nuovo Cimento 12, 6 (1989) 1-202. FORMULA 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 EXAMPLE a(8) = 7 - 4 = 3; the corresponding partitions are 44, 332 and 2222. 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 *) PROG (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 CROSSREFS Cf. A000041, A002865, A072380, A081094, A081095. Sequence in context: A263352 A008734 A226649 * A175990 A290982 A162517 Adjacent sequences:  A053442 A053443 A053444 * A053446 A053447 A053448 KEYWORD easy,nice,nonn AUTHOR Alford Arnold, Jan 12 2000 EXTENSIONS More terms from James A. Sellers, Feb 02 2000 Start of sequence changed, Apr 25 2003 STATUS approved

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Last modified October 21 03:24 EDT 2019. Contains 328291 sequences. (Running on oeis4.)