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A232605 Number of compositions of 2n into parts with multiplicity <= n.

%I A232605 #21 Mar 31 2017 03:51:46
%S A232605 1,1,7,26,114,459,1892,7660,31081,125464,506025,2036706,8189555,
%T A232605 32894825,132033140,529614616,2123365038,8509634259,34092146068,
%U A232605 136546197412,546774790297,2189060331762,8762770476060,35072837719356,140363923730474,561697985182654
%N A232605 Number of compositions of 2n into parts with multiplicity <= n.
%C A232605 a(n) = A243081(2n,n) = Sum_{i=0..n} A242447(2n,i).
%H A232605 Alois P. Heinz, <a href="/A232605/b232605.txt">Table of n, a(n) for n = 0..1000</a>
%F A232605 Recurrence: 5*(n-2)*(n-1)*n*(1258*n^4 - 11230*n^3 + 37013*n^2 - 53645*n + 28764)*a(n) = 2*(n-2)*(n-1)*(17612*n^5 - 159736*n^4 + 538872*n^3 - 824111*n^2 + 541051*n - 107568)*a(n-1) - 4*(n-2)*(5032*n^5 - 44925*n^4 + 134332*n^3 - 137541*n^2 - 6614*n + 52596)*a(n-2) - 2*(83028*n^7 - 1074550*n^6 + 5758938*n^5 - 16516699*n^4 + 27297714*n^3 - 25934731*n^2 + 13070460*n - 2661120)*a(n-3) + 8*(n-4)*(n-1)*(2*n-7)*(1258*n^4 - 6198*n^3 + 10871*n^2 - 8277*n + 2160)*a(n-4). - _Vaclav Kotesovec_, Nov 27 2013
%F A232605 a(n) ~ 2^(2*n-1). - _Vaclav Kotesovec_, Nov 27 2013
%e A232605 a(1) = 1: [2].
%e A232605 a(2) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
%e A232605 a(3) = 26: [6], [5,1], [4,2], [3,3], [2,4], [1,5], [3,2,1], [2,3,1], [1,4,1], [3,1,2], [2,2,2], [1,3,2], [2,1,3], [1,2,3], [1,1,4], [4,1,1], [2,1,2,1], [1,2,2,1], [1,1,3,1], [3,1,1,1], [2,2,1,1], [1,1,2,2], [1,1,1,3], [1,3,1,1], [2,1,1,2], [1,2,1,2].
%p A232605 a:= proc(n) option remember;
%p A232605      `if`(n<5, [1, 1, 7, 26, 114][n+1],
%p A232605       (2*(n-1)*(11092322562903*n^3 -66692687083623*n^2
%p A232605        +117736395568913*n -51473509383358) *a(n-1)
%p A232605       -(17386283060104*n^4 -178154697569624*n^3 +652039987731328*n^2
%p A232605        -984836231488344*n +485931992440304) *a(n-2)
%p A232605       -(89948343833304*n^4 -664733317200192*n^3 +1662507315916082*n^2
%p A232605        -1594206267597886*n +485625773146800) *a(n-3)
%p A232605       +(92866735410328*n^4 -1047423564207444*n^3 +4160804083968884*n^2
%p A232605        -6634447008138888*n +3217864137236880) *a(n-4)
%p A232605       -16*(n-5)*(2*n-9)*(310469340359*n^2 -847919784312*n
%p A232605        +494768703748) *a(n-5)) / (5*n*(n-1)*
%p A232605       (681426847222*n^2 -3587414825361*n +4663189129034)))
%p A232605     end:
%p A232605 seq(a(n), n=0..40);
%t A232605 b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j, k]/j!, {j, 0, Min[n/i, k]}]]];
%t A232605 A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n - 1)], b[n, n, 0, k]];
%t A232605 a[n_] := A[2 n, n];
%t A232605 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 31 2017, after _Alois P. Heinz_ *)
%Y A232605 Cf. A232623, A232665.
%K A232605 nonn
%O A232605 0,3
%A A232605 _Alois P. Heinz_, Nov 26 2013