A348474 - OEIS

%I #20 Apr 07 2022 10:46:43

%S 1,2,8,41,220,1212,6803,38691,222196,1285610,7482718,43762754,

%T 256972507,1514020484,8945944435,52990732161,314568593860,

%U 1870939233546,11146516959176,66508200091575,397375460647690,2377167144881136,14236462650026064,85346464443885086

%N Number of compositions of n into exactly 2n nonnegative parts such that each positive i-th part is odd if i is odd.

%H Alois P. Heinz, <a href="/A348474/b348474.txt">Table of n, a(n) for n = 0..1272</a>

%F a(n) ~ c * d^n / sqrt(Pi*n), where d = 6.12846447590595003785095345916525... is the real root of the equation 32*d^4 - 195*d^3 + 12*d^2 - 112*d - 20 = 0 and c = 0.5667463795063214394117147185755881... is positive root of the equation 182464*c^8 - 45616*c^6 - 2108*c^4 - 601*c^2 - 20 = 0. - _Vaclav Kotesovec_, Nov 01 2021

%F From _Peter Bala_, Feb 22 2022: (Start)

%F Conjecture: a(n) = [x^n] ( (1 + x - x^2)/((1 + x)*(1 - x)^2) )^n.

%F If true, then the following hold:

%F a(n) = Sum_{i = 0..n} Sum_{j = 0..n} binomial(n,i+2*j)*binomial(2*i+2*j-1, i)*binomial(n+j-1,j).

%F exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 6*x^2 + 23*x^3 + 99*x^4 + ... is the g.f. of A133656.

%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

%e a(2) = 8: [0,0,0,2], [0,0,1,1], [0,1,0,1], [0,1,1,0], [0,2,0,0], [1,0,0,1], [1,0,1,0], [1,1,0,0].

%p b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),

%p       add(b(n-j, t-1)*iquo(j+3, 2), j=0..n))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..25);

%t b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],

%t      Sum[b[n - j, t - 1]*Quotient[j + 3, 2], {j, 0, n}]];

%t a[n_] := b[n, n];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 07 2022, after _Alois P. Heinz_ *)

%Y Cf. A133656, A348410, A348476, A348478.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 19 2021