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A332051 Number of compositions of 2n where the multiplicity of the first part equals n.

%I A332051 #22 Dec 20 2020 12:20:19
%S A332051 1,1,3,4,15,36,126,372,1239,3910,12848,41581,136578,447188,1473342,
%T A332051 4855704,16053831,53138244,176233968,585202262,1945964080,6478043121,
%U A332051 21588979877,72016891509,240452892570,803489258286,2686964354376,8991840800137,30110638705890
%N A332051 Number of compositions of 2n where the multiplicity of the first part equals n.
%H A332051 Alois P. Heinz, <a href="/A332051/b332051.txt">Table of n, a(n) for n = 0..1882</a>
%F A332051 a(n) = A331332(2n,n).
%F A332051 a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.40869819984215108586487649733361214893... is the root of the equation 4 - 32*d - 8*d^2 + 5*d^3 = 0, and c = 0.34930509632919368540449993196290415079... is the root of the equation 5 - 4*c^2 - 592*c^4 + 2368*c^6 = 0. - _Vaclav Kotesovec_, Feb 08 2020
%F A332051 Recurrence: 5*(n-1)*n*(2294*n^5 - 31267*n^4 + 168064*n^3 - 445121*n^2 + 580494*n - 297864)*a(n) = (n-1)*(29822*n^6 - 415647*n^5 + 2327634*n^4 - 6668807*n^3 + 10238782*n^2 - 7910608*n + 2368800)*a(n-1) + 2*(27528*n^7 - 434848*n^6 + 2851985*n^5 - 10024036*n^4 + 20278349*n^3 - 23438626*n^2 + 14189888*n - 3420000)*a(n-2) - 2*(41292*n^7 - 647684*n^6 + 4218357*n^5 - 14743832*n^4 + 29759871*n^3 - 34533464*n^2 + 21199620*n - 5259600)*a(n-3) + 2*(n-4)*(2*n - 7)*(2294*n^5 - 19797*n^4 + 65936*n^3 - 105591*n^2 + 80846*n - 23400)*a(n-4). - _Vaclav Kotesovec_, Feb 08 2020
%e A332051 a(0) = 1: the empty composition.
%e A332051 a(1) = 1: 2.
%e A332051 a(2) = 3: 22, 112, 121.
%e A332051 a(3) = 4: 222, 1113, 1131, 1311.
%e A332051 a(4) = 15: 2222, 11114, 11141, 11411, 14111, 111122, 111212, 111221, 112112, 112121, 112211, 121112, 121121, 121211, 122111.
%p A332051 b:= proc(n, i) option remember; `if`(n=0, x, add(expand(
%p A332051      `if`(i=j, x, 1)*b(n-j, `if`(n<i+j, 0, i))), j=1..n))
%p A332051     end:
%p A332051 a:= n-> `if`(n=0, 1, coeff(add(b(2*n-j, j), j=1..2*n), x, n)):
%p A332051 seq(a(n), n=0..35);
%t A332051 b[n_, i_] := b[n, i] = If[n == 0, x, Sum[Expand[If[i == j, x, 1] b[n - j, If[n < i + j, 0, i]]], {j, 1, n}]];
%t A332051 a[n_] := If[n == 0, 1, Coefficient[Sum[b[2 n - j, j], {j, 1, 2 n}], x, n]];
%t A332051 a /@ Range[0, 35] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)
%Y A332051 Cf. A232665, A232697, A255197, A331332.
%K A332051 nonn
%O A332051 0,3
%A A332051 _Alois P. Heinz_, Feb 06 2020