A186997 - OEIS

%I #83 Nov 22 2024 09:22:49

%S 1,1,4,19,104,614,3816,24595,162896,1101922,7580904,52878654,

%T 373100272,2658188524,19096607120,138182654595,1006202473888,

%U 7367648586954,54214472633064,400698865376842,2973344993337520,22142778865313364

%N G.f. satisfies: A(x) = 1 + x*A(x)^3 + x^2*A(x)^4.

%C a(n-1) is the number of rows with the value false in the truth tables of all bracketed implications with n distinct variables. - _Volkan Yildiz_, Jul 01 2011

%C From _Peter Bala_, Aug 02 2016: (Start)

%C Conjectures (both checked up to n = 100):

%C 2-adic valuation of a(2*n+1) = 2-adic valuation of Catalan(2*n+1) (= 2-adic valuation of Catalan(n));

%C 2-adic valuation of a(2*n) = 3 + 2-adic valuation of Catalan(n-2) for n >= 2, where Catalan(n) = A000108(n). (End)

%C a(n) is the number of Dyck paths of semi-length n using only steps U_1 = (4,2), U_2 = (9,3) and D = (1,-1). - _Michael D. Weiner_, Jun 12 2017

%H Vincenzo Librandi, <a href="/A186997/b186997.txt">Table of n, a(n) for n = 0..100</a>

%H Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara and Michael D. Weiner, <a href="https://arxiv.org/abs/1602.03550">Enumeration of colored Dyck paths via partial Bell polynomials</a>, arXiv:1602.03550 [math.CO], 2016.

%H P. J. Cameron and V. Yildiz, <a href="http://www.maths.qmul.ac.uk/~pjc/preprints/asim2.pdf">Counting false entries in truth tables of bracketed formulas connected by implication</a>. Also arXiv:<a href="https://arxiv.org/abs/1106.4443">1106.4443</a> [math.CO], 2011.

%H Volkan Yildiz, <a href="https://arxiv.org/abs/1205.5595">General combinatorical structure of truth tables of bracketed formulas connected by implication</a>, arXiv preprint arXiv:1205.5595 [math.CO], 2012.

%H Volkan Yildiz, <a href="https://arxiv.org/abs/2106.04728">Notes on algebraic structure of truth tables of bracketed formulae connected by implications</a>, arXiv:2106.04728 [math.CO], 2021. See S(x) p. 3.

%F G.f.: A(x) = (1/x)*Series_Reversion(x*(1+sqrt(1-4*x-4*x^2))/2).

%F a(n) = Sum_{k=0..n} binomial(k,n-k)*binomial(n+2*k,n+k)/(n+k+1). - _Vladimir Kruchinin_, May 12 2011

%F From _Volkan Yildiz_, Jul 03 2011: (Start)

%F Let f(n) = Sum_{i=1..n-1} (2^i*C(i-1)-f(i))*f(n-i), with f(0)=0, f(1)=1, and where C are the Catalan numbers A000108. Then a(n)=f(n+1).

%F G.f.: (1/x) * (-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/4.

%F For large n, a(n) is asymptotically (3-sqrt(3))/3* 2^(3*n)/sqrt(Pi*n^3), corrected by _Vaclav Kotesovec_, May 31 2014. (End)

%F O.g.f. satisfies A(x^2) = 1/x * Series_Reversion( x*(1 - x^2)/(1 + x^4) ). - _Peter Bala_, Aug 02 2016

%F D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -n*(28*n^2-27*n+8)*a(n-1) +6*(11*n^3-36*n^2+32*n-5)*a(n-2) +8*(2*n-5)*(14*n^2-52*n+51)*a(n-3) +32*(2*n-5)*(2*n-7)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Nov 22 2024

%e G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 104*x^4 + 614*x^5 + 3816*x^6 + ...

%p A(x):= 1/x * (-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/4 ;

%p a:= n-> coeff(series(A(x), x, n+2), x, n):

%p seq(a(n), n=0..20); # _Volkan Yildiz_, Jul 01 2011

%t CoefficientList[Series[1/x * (-1-Sqrt[1-8*x]+Sqrt[2+2*Sqrt[1-8*x]+8*x])/4, {x, 0, 20}], x] (* _Vaclav Kotesovec_, May 31 2014 *)

%t a[n_] := Sum[Binomial[k, n-k]*Binomial[n+2*k, n+k]/(n+k+1), {k, 1, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 02 2015, after _Vladimir Kruchinin_ *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^3+x^2*(A+x*O(x^n))^4);polcoeff(A,n)}

%o for(n=0,25,print1(a(n),", "))

%o (PARI) {a(n)=polcoeff((1/x)*serreverse(x*(1+sqrt(1-4*x-4*x^2 +x^2*O(x^n)))/2),n)}

%o for(n=0,25,print1(a(n),", "))

%o (Maxima) a(n):=sum((binomial(k,n-k)*binomial(n+2*k,n+k))/(n+k+1),k,1,n); /* _Vladimir Kruchinin_, May 12 2011 */

%o (PARI) x='x+O('x^66); /* that many terms */

%o Vec(1/x*serreverse((x*(1+sqrt(1-4*x-4*x^2))/2))) /* show terms */ /* _Joerg Arndt_, May 13 2011 */

%Y Cf. A000108.

%K nonn,easy

%O 0,3

%A _Paul D. Hanna_, Mar 01 2011