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%I #121 Sep 08 2022 08:44:49
%S 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,
%T 136,150,169,185,206,225,249,270,297,321,351,378,411,441,478,511,551,
%U 588,632,672,720,764,816,864,920,972,1033,1089,1154,1215,1285,1350
%N Number of partitions of n in which the greatest part is 4.
%C Also number of partitions of n into exactly 4 parts.
%C Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - _R. J. Mathar_, Nov 03 2018
%C From _Gus Wiseman_, Jun 27 2021: (Start)
%C 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:
%C (4) (5,1) (6,2) (7,3) (8,4) (9,5) (10,6)
%C (5,2,1) (5,3,2) (5,4,3) (6,5,3) (7,6,3)
%C (6,3,1) (6,4,2) (7,5,2) (8,6,2)
%C (7,4,1) (8,5,1) (9,6,1)
%C (6,3,2,1) (6,4,3,1) (6,5,4,1)
%C (7,4,2,1) (7,4,3,2)
%C (7,5,3,1)
%C (8,5,2,1)
%C (6,4,3,2,1)
%C (End)
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
%D 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.
%H Washington Bomfim, <a href="/A026810/b026810.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-2,0,0,1,1,-1).
%F 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)).
%F a(n+4) = A001400(n). - _Michael Somos_, Apr 07 2012
%F a(n) = round( (n^3 + 3*n^2 -9*n*(n mod 2))/144 ). - _Washington Bomfim_, Jan 06 2021 and Jul 03 2012
%F a(n) = (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 -A049347(n)/9 +A056594(n)/8. - _R. J. Mathar_, Jul 03 2012
%F From _Gregory L. Simay_, Oct 13 2015: (Start)
%F 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.
%F 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)
%F a(n) = A008284(n,4). - _Robert A. Russell_, May 13 2018
%F From _Gregory L. Simay_, Jul 28 2019: (Start)
%F a(2n+1) = a(2n) + a(n+1) - a(n-3) and
%F a(2n) = a(2n-1) + a(n+2) - a(n-2). (End)
%e From _Gus Wiseman_, Jun 27 2021: (Start)
%e The a(4) = 1 through a(10) = 9 partitions of length 4:
%e (1111) (2111) (2211) (2221) (2222) (3222) (3322)
%e (3111) (3211) (3221) (3321) (3331)
%e (4111) (3311) (4221) (4222)
%e (4211) (4311) (4321)
%e (5111) (5211) (4411)
%e (6111) (5221)
%e (5311)
%e (6211)
%e (7111)
%e (End)
%p A049347 := proc(n)
%p op(1+(n mod 3),[1,-1,0]) ;
%p end proc:
%p A056594 := proc(n)
%p op(1+(n mod 4),[1,0,-1,0]) ;
%p end proc:
%p A026810 := proc(n)
%p 1/288*(n+1)*(2*n^2+4*n-13+9*(-1)^n) ;
%p %-A049347(n)/9 ;
%p %+A056594(n)/8 ;
%p end proc: # _R. J. Mathar_, Jul 03 2012
%t Table[Count[IntegerPartitions[n], {4, ___}], {n, 0, 60}]
%t 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 *)
%t Table[Length[IntegerPartitions[n, {4}]], {n, 0, 60}] (* _Eric Rowland_, Mar 02 2017 *)
%t CoefficientList[Series[x^4/Product[1 - x^k, {k, 1, 4}], {x, 0, 60}], x] (* _Robert A. Russell_, May 13 2018 *)
%o (PARI) for(n=0, 60, print(n, " ", round((n^3 + 3*n^2 -9*n*(n % 2))/144))); \\ _Washington Bomfim_, Jul 03 2012
%o (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
%o (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
%o (PARI) print1(0,", "); for(n=1,60,j=0;forpart(v=n,j++,,[4,4]); print1(j,", ")) \\ _Hugo Pfoertner_, Oct 01 2018
%o (Magma) [Round((n^3+3*n^2-9*n*(n mod 2))/144): n in [0..60]]; // _Vincenzo Librandi_, Oct 14 2015
%Y Cf. A001400, A026811, A026812, A026813, A026814, A026815, A026816, A069905 (3 positive parts), A002621 (partial sums), A005044 (first differences).
%Y A non-strict version is A000710 or A088218.
%Y This is column k = 2 of A152146.
%Y A reverse version is A343941.
%Y Cf. A000041, A000070, A000097, A067659, A103919, A120452, A236559, A239830, A306145, A343942, A344616, A344649, A344651.
%K nonn,easy
%O 0,7
%A _Clark Kimberling_
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