A348474 Number of compositions of n into exactly 2n nonnegative parts such that each positive i-th part is odd if i is odd.
1, 2, 8, 41, 220, 1212, 6803, 38691, 222196, 1285610, 7482718, 43762754, 256972507, 1514020484, 8945944435, 52990732161, 314568593860, 1870939233546, 11146516959176, 66508200091575, 397375460647690, 2377167144881136, 14236462650026064, 85346464443885086
Offset: 0
Keywords
Examples
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].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1272
Programs
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Maple
b:= proc(n, t) option remember; `if`(t=0, 1-signum(n), add(b(n-j, t-1)*iquo(j+3, 2), j=0..n)) end: a:= n-> b(n$2): seq(a(n), n=0..25); -
Mathematica
b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n], Sum[b[n - j, t - 1]*Quotient[j + 3, 2], {j, 0, n}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 07 2022, after Alois P. Heinz *)
Formula
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
From Peter Bala, Feb 22 2022: (Start)
Conjecture: a(n) = [x^n] ( (1 + x - x^2)/((1 + x)*(1 - x)^2) )^n.
If true, then the following hold:
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).
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.
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)