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A348474 Number of compositions of n into exactly 2n nonnegative parts such that each positive i-th part is odd if i is odd.

%I A348474 #20 Apr 07 2022 10:46:43
%S A348474 1,2,8,41,220,1212,6803,38691,222196,1285610,7482718,43762754,
%T A348474 256972507,1514020484,8945944435,52990732161,314568593860,
%U A348474 1870939233546,11146516959176,66508200091575,397375460647690,2377167144881136,14236462650026064,85346464443885086
%N A348474 Number of compositions of n into exactly 2n nonnegative parts such that each positive i-th part is odd if i is odd.
%H A348474 Alois P. Heinz, <a href="/A348474/b348474.txt">Table of n, a(n) for n = 0..1272</a>
%F A348474 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 A348474 From _Peter Bala_, Feb 22 2022: (Start)
%F A348474 Conjecture: a(n) = [x^n] ( (1 + x - x^2)/((1 + x)*(1 - x)^2) )^n.
%F A348474 If true, then the following hold:
%F A348474 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 A348474 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 A348474 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 A348474 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 A348474 b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
%p A348474       add(b(n-j, t-1)*iquo(j+3, 2), j=0..n))
%p A348474     end:
%p A348474 a:= n-> b(n$2):
%p A348474 seq(a(n), n=0..25);
%t A348474 b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
%t A348474      Sum[b[n - j, t - 1]*Quotient[j + 3, 2], {j, 0, n}]];
%t A348474 a[n_] := b[n, n];
%t A348474 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 07 2022, after _Alois P. Heinz_ *)
%Y A348474 Cf. A133656, A348410, A348476, A348478.
%K A348474 nonn
%O A348474 0,2
%A A348474 _Alois P. Heinz_, Oct 19 2021