|
%I #10 Oct 31 2018 17:02:34
%S 0,1,5448,517971,10388788,97809616,587267282,2615047418,9408671330,
%T 28851490163,78132541528,191563698893,432971108530,914435915008,
%U 1823570327812,3461969170632,6297974185728,11037428544793,18716682230144,30822735206253,49446074607790
%N Number of 10-element subsets that can be chosen from {1,2,...,10*n} having element sum 50*n+5.
%C a(n) is the number of partitions of 50*n+5 into 10 distinct parts <= 10*n.
%H Alois P. Heinz, <a href="/A204474/b204474.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (2934*x^31 +348266*x^30 +6597024*x^29 +49734090*x^28 +217728951*x^27 +667463008*x^26 +1585348242*x^25 +3103463874*x^24 +5228487221*x^23 +7832917054*x^22 +10700328447*x^21 +13574603633*x^20 +16172892190*x^19 +18185529447*x^18 +19316679453*x^17 +19372433191*x^16 +18342644114*x^15 +16405735056*x^14 +13851791136*x^13 +10991882803*x^12 +8111849599*x^11 +5470428985*x^10 +3289797304*x^9 +1708913237*x^8 +735273835*x^7 +246947710*x^6 +58833425*x^5 +8344142*x^4 +496184*x^3 +5444*x^2+x) / ((x^2+1)*(x^6+x^3+1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x+1)^2*(x^2+x+1)^3*(x-1)^10).
%e a(2) = 5448 because there are 5448 10-element subsets that can be chosen from {1,2,...,20} having element sum 105: {1,2,3,4,5,16,17,18,19,20},{1,2,3,4,6,15,17,18,19,20}, ..., {6,7,8,9,10,11,12,13,14,15}.
%p a:= n-> (Matrix(32, (i, j)-> `if`(i=j-1, 1, `if`(i=32, [-1, 4, -5, 2, -2, 2, 5, -7, 2, -2, -1, 9, -6, -1, 1, -7, 14, -7, 1, -1, -6, 9, -1, -2, 2, -7, 5, 2, -2, 2, -5, 4][j], 0)))^n.
%p <<0, 1, 5448, 517971, 10388788, 97809616, 587267282, 2615047418, 9408671330, 28851490163, 78132541528, 191563698893, 432971108530, 914435915008, 1823570327812, 3461969170632, 6297974185728, 11037428544793, 18716682230144,
%p 30822735206253, 49446074607790, 77472474506552, 118820782200294, 178734513401148, 264135920159882, 384052079868415, 550123481245484, 777206553792057, 1084082600833512, 1494286652957642, 2037070858360564, 2748518169368740>>)[1, 1]: seq(a(n), n=0..50);
%Y Column k=10 of A204459.
%K nonn,easy
%O 0,3
%A _Alois P. Heinz_, Jan 16 2012
|
