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A026810
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Number of partitions of n in which the greatest part is 4.
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40
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0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350
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OFFSET
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0,7
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COMMENTS
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Also number of partitions of n into exactly 4 parts.
Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - R. J. Mathar, Nov 03 2018
Also the number of strict integer partitions of 2n with alternating sum 4, or (by conjugation) partitions of 2n covering an initial interval of positive integers with exactly 4 odd parts. The strict partitions with alternating sum 4 are:
(4) (5,1) (6,2) (7,3) (8,4) (9,5) (10,6)
(5,2,1) (5,3,2) (5,4,3) (6,5,3) (7,6,3)
(6,3,1) (6,4,2) (7,5,2) (8,6,2)
(7,4,1) (8,5,1) (9,6,1)
(6,3,2,1) (6,4,3,1) (6,5,4,1)
(7,4,2,1) (7,4,3,2)
(7,5,3,1)
(8,5,2,1)
(6,4,3,2,1)
(End)
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
D. E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).
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FORMULA
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G.f.: x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) = x^4/((1-x)^4*(1+x)^2*(1+x+x^2)*(1+x^2)).
a(n) = round( (n^3 + 3*n^2 -9*n*(n mod 2))/144 ). - Washington Bomfim, Jan 06 2021 and Jul 03 2012
a(n) = (n^3 + 3*n^2 - 9*n)/144 + a(m) - (m^3 + 3*m^2 - 9*m)/144 if n = 12k + m and m is odd. For example, a(23) = a(12*1 + 11) = (23^3 + 3*23^2 - 9*23)/144 + a(11) - (11^3 + 3*11^2 - 9*11)/144 = 94.
a(n) = (n^3 + 3*n^2)/144 + a(m) - (m^3 + 3*m^2)/144 if n = 12k + m and m is even. For example, a(22) = a(12*1 + 10) = (22^3 + 3*22^2)/144 + a(10) - (10^3 + 3*10^2)/144 = 84. (End)
a(2n+1) = a(2n) + a(n+1) - a(n-3) and
a(2n) = a(2n-1) + a(n+2) - a(n-2). (End)
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EXAMPLE
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The a(4) = 1 through a(10) = 9 partitions of length 4:
(1111) (2111) (2211) (2221) (2222) (3222) (3322)
(3111) (3211) (3221) (3321) (3331)
(4111) (3311) (4221) (4222)
(4211) (4311) (4321)
(5111) (5211) (4411)
(6111) (5221)
(5311)
(6211)
(7111)
(End)
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MAPLE
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op(1+(n mod 3), [1, -1, 0]) ;
end proc:
op(1+(n mod 4), [1, 0, -1, 0]) ;
end proc:
1/288*(n+1)*(2*n^2+4*n-13+9*(-1)^n) ;
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MATHEMATICA
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Table[Count[IntegerPartitions[n], {4, ___}], {n, 0, 60}]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {0, 0, 0, 0, 1, 1, 2, 3, 5, 6}, 60] (* Vincenzo Librandi, Oct 14 2015 *)
Table[Length[IntegerPartitions[n, {4}]], {n, 0, 60}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^4/Product[1 - x^k, {k, 1, 4}], {x, 0, 60}], x] (* Robert A. Russell, May 13 2018 *)
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PROG
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(PARI) for(n=0, 60, print(n, " ", round((n^3 + 3*n^2 -9*n*(n % 2))/144))); \\ Washington Bomfim, Jul 03 2012
(PARI) x='x+O('x^60); concat([0, 0, 0, 0], Vec(x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)))) \\ Altug Alkan, Oct 14 2015
(PARI) vector(60, n, n--; (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 + real(I^n)/8 - ((n+2)%3-1)/9) \\ Altug Alkan, Oct 26 2015
(PARI) print1(0, ", "); for(n=1, 60, j=0; forpart(v=n, j++, , [4, 4]); print1(j, ", ")) \\ Hugo Pfoertner, Oct 01 2018
(Magma) [Round((n^3+3*n^2-9*n*(n mod 2))/144): n in [0..60]]; // Vincenzo Librandi, Oct 14 2015
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CROSSREFS
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Cf. A000041, A000070, A000097, A067659, A103919, A120452, A236559, A239830, A306145, A343942, A344616, A344649, A344651.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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