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A026810 Number of partitions of n in which the greatest part is 4. 24

%I

%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

%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+3n^2-9n[n odd])/144, where [...] denotes the Iverson bracket [true] = 1 and [false] = 0. - _Washington Bomfim_, 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

%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}, 80] (* _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, 65}], x] (* _Robert A. Russell_, May 13 2018 *)

%o (PARI)

%o for(n=0, 10000, print(n, " ", round((n^3 + 3*n^2 -9*n*(n % 2))/144)));

%o \\ _Washington Bomfim_, Jul 03 2012

%o (MAGMA) [Round((n^3+3*n^2-9*n*(n mod 2))/144): n in [0..80]]; // _Vincenzo Librandi_, Oct 14 2015

%o (PARI) x='x+O('x^100); 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(100, 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=1,57,j=0;forpart(v=n,j++,,[4,4]);print1(j,", ")) \\ _Hugo Pfoertner_, Oct 01 2018

%Y Cf. A001400, A026811, A026812, A026813, A026814, A026815, A026816, A069905 (3 positive parts), A002621 (partial sums), A005044 (first differences).

%K nonn,easy

%O 0,7

%A _Clark Kimberling_

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Last modified January 21 19:54 EST 2019. Contains 319350 sequences. (Running on oeis4.)