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%I #63 Jul 30 2022 12:03:56
%S 1,1,2,3,5,7,11,15,21,28,38,49,65,82,105,131,164,201,248,300,364,436,
%T 522,618,733,860,1009,1175,1367,1579,1824,2093,2400,2738,3120,3539,
%U 4011,4526,5102,5731,6430,7190,8033,8946,9953,11044,12241,13534,14950,16475,18138
%N Number of partitions of n into at most 7 parts.
%C Also, the number of partitions of n into parts <= 7: a(n) = A026820(n, 7). - _Reinhard Zumkeller_, Jan 21 2010
%C Counts unordered closed walks of weight n on a single vertex graph with 7 loops of weights 1, 2, 3, 4, 5, 6 and 7. - _David Neil McGrath_, Apr 11 2015
%C Number of different distributions of n+28 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - _Ece Uslu_ and Esin Becenen, Jan 11 2016
%D A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%H T. D. Noe, <a href="/A008636/b008636.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=356">Encyclopedia of Combinatorial Structures 356</a>
%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,0,-1,-1,0,1,1,2,0,0,0,-2,-1,-1,0,1,1,0,1,0,0,-1,-1,1).
%F G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).
%F a(n) = A008284(n+7, 7), n >= 0.
%F a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28). - _David Neil McGrath_, Apr 11 2015
%F a(n+7) = a(n) + A001402(n). - _Ece Uslu_, Esin Becenen, Jan 11 2016
%F a(n) = A026813(n+7). - _R. J. Mathar_, Feb 13 2019
%F From _Vladimír Modrák_, Jul 30 2022: (Start)
%F a(n) = Sum_{p=0..floor(n/7)} Sum_{m=0..floor(n/6)} Sum_{k=0..floor(n/5)} Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0, n + 1 - 3*i - 4*j - 5*k - 6*m - 7*p))/2).
%F a(n) = Sum_{m=0..floor(n/7)} Sum_{k=0..floor(n/6)} Sum_{j=0..floor(n/5)} Sum_{i=0..floor(n/4)} floor(((max(0, n + 3 - 4*i - 5*j - 6*k - 7*m))^2+4)/12). (End)
%e There are 28 partitions of 9 into parts less than or equal to 7. These are (72)(711)(63)(621)(6111)(54)(531)(522)(5211)(51111)(441)(432)(4311)(4221)(42111)(411111)(333)(3321)(33111)(3222)(32211)(321111)(3111111)(22221)(222111)(2211111)(21111111)(111111111). - _David Neil McGrath_, Apr 11 2015
%e a(3) = 3, i.e., {1,2,3,4,5,7,9}, {1,2,3,4,6,7,8}, {1,2,3,4,5,6,10}. Number of different distributions of 31 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - _Ece Uslu_, Esin Becenen, Jan 11 2016
%p with(combstruct):ZL8:=[S,{S=Set(Cycle(Z,card<8))}, unlabeled]: seq(count(ZL8,size=n),n=0..48); # _Zerinvary Lajos_, Sep 24 2007
%p B:=[S,{S = Set(Sequence(Z,1 <= card),card <=7)},unlabelled]: seq(combstruct[count](B, size=n), n=0..48); # _Zerinvary Lajos_, Mar 21 2009
%t CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 7} ], {x, 0, 60} ], x ]
%o (PARI) {a(n)=(2*n^6+168*n^5+5530*n^4+90160*n^3+754299*n^2+(2988020+44800*(1-n%3))*n+6654375+1575*(3*n^2+84*n+511)*(-1)^n)\7257600}; \\ _Tani Akinari_, May 27 2014
%Y Cf. A008284, A026820.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Mar 15 1996
%E More terms from _Robert G. Wilson v_, Dec 11 2000
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