A186241 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*A(x)^4 + x^3*A(x)^6.

1, 1, 3, 12, 54, 262, 1337, 7072, 38426, 213197, 1202795, 6879160, 39794416, 232429030, 1368806610, 8118934656, 48458809586, 290832756606, 1754059333738, 10625545472716, 64620970743082, 394409682103262, 2415084675723048, 14832185219521152, 91339478577683664

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1108

Nathan Gabriel, Katherine Peske, Lara Pudwell, and Samuel Tay, Pattern avoidance in ternary trees J. Integer Seq. 15 (2012), no. 1, Article 12.1.5, 20 pp.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

a(n) = 1/(2*n-1)*Sum_{j=0..2*n-1} binomial(2*n-1,j)*Sum_{i=j..n+j-1} binomial(j,i-j)*binomial(2*n-j-1,3*j-3*n-i+1))), n>0.

From Paul D. Hanna, Nov 11 2011: (Start)

G.f. A(x) satisfies:

(1) A(x) = sqrt( (1/x)*Series_Reversion( x/(1 + x + x^2 + x^3)^2 ) ).

(2) A( x/(1 + x + x^2 + x^3)^2 ) = 1 + x + x^2 + x^3.

(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = g.f. of A036765 (number of rooted trees with a degree constraint).

(4) a(n) = [x^n] (1 + x + x^2 + x^3)^(2*n+1) / (2*n+1).

(5) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * [Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(2*k)] ).

(6) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * [(1-x*A(x)^2)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^(2*k) )] ).

(End)

From Peter Bala, Jun 21 2015: (Start)

a(n) = 1/(2*n + 1)*Sum_{k = 0..floor(n/2)} binomial(2*n + 1,k)*binomial(2*n + 1,n - 2*k).

More generally, the coefficient of x^n in A(x)^r equals r/(2*n + r)*Sum_{k = 0..floor(n/2)} binomial(2*n + r,k)*binomial(2*n + r,n - 2*k) by the Lagrange-Bürmann formula.

O.g.f. A(x) = exp(Sum_{n >= 1} 1/2*b(n)*x^n/n), where b(n) = Sum_{k = 0..floor(n/2)} binomial(2*n,k)*binomial(2*n,n - 2*k). Cf. A036765, A198951, A200731. (End)

Recurrence: 5*n*(5*n - 1)*(5*n + 1)*(5*n + 2)*(5*n + 3)*(13144*n^4 - 57784*n^3 + 90149*n^2 - 59354*n + 13980)*a(n) = 8*(2*n - 1)*(16259128*n^8 - 71478808*n^7 + 108653137*n^6 - 60530902*n^5 - 2811173*n^4 + 12694433*n^3 - 2398482*n^2 - 352503*n + 78570)*a(n-1) + 128*(n-1)*(2*n - 3)*(2*n - 1)*(52576*n^6 - 178560*n^5 + 136156*n^4 + 22938*n^3 - 16067*n^2 - 3138*n - 405)*a(n-2) + 2048*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(13144*n^4 - 5208*n^3 - 4339*n^2 + 168*n + 135)*a(n-3). - Vaclav Kotesovec, Nov 17 2017

A(x^2) = (1/x) * series reversion of x/(1 + x^2 + x^4 + x^6). - Peter Bala, Jul 27 2023

Example

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 54*x^4 + 262*x^5 + 1337*x^6 +...
where A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)^4).
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 141*x^4 + 704*x^5 + 3666*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 88*x^3 + 451*x^4 + 2392*x^5 + 13022*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 182*x^3 + 1014*x^4 + 5718*x^5 + 32623*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x)^4 + x^3*A(x)^6.
From Paul D. Hanna, Nov 11 2011: (Start)
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A^2)*x*A + (1 + 2^2*x*A^2 + x^2*A^4)*x^2*A^2/2 +
(1 + 3^2*x*A^2 + 3^2*x^2*A^4 + x^3*A^6)*x^3*A^3/3 +
(1 + 4^2*x*A^2 + 6^2*x^2*A^4 + 4^2*x^3*A^6 + x^4*A^8)*x^4*A^4/4 +
(1 + 5^2*x*A^2 + 10^2*x^2*A^4 + 10^2*x^3*A^6 + 5^2*x^4*A^8 + x^5*A^10)*x^5*A^5/5 + ...
which involves squares of binomial coefficients. (End)

Maple

F:= proc(n) if n::even then
  simplify((1/2)*hypergeom([-(1/2)*n, -2*n-1, -(1/2)*n+1/2], [(1/2)*n+1, 3/2+(1/2)*n], -1)*(2*n+2)!/((2*n+1)*((n+1)!)^2))
  else
  simplify((1/2)*hypergeom([-(1/2)*n, -2*n-1, -(1/2)*n+1/2], [(1/2)*n+1, 3/2+(1/2)*n], -1)*(2*n+2)!/((2*n+1)*((n+1)!)^2))
  fi
end proc:
map(F, [$0..30]); # Robert Israel, Jun 22 2015

Mathematica

a[n_] := 1/(2n + 1) Sum[Binomial[2n + 1, k] Binomial[2n + 1, n - 2k], {k, 0, n/2}];
(* or: *)
a[n_] := (Binomial[2n + 1, n] HypergeometricPFQ[{-2n - 1, 1/2 - n/2, -n/2}, {n/2 + 1, n/2 + 3/2}, -1])/(2n + 1);
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 17 2017 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A^2)*(1+x^2*A^4)+x*O(x^n)); polcoeff(A, n)} /* Paul D. Hanna */
(PARI) {a(n)=polcoeff(sqrt((1/x)*serreverse(x/(1 + x + x^2 + x^3 +x*O(x^n))^2)), n)} /* Paul D. Hanna */
(PARI) {a(n)=polcoeff( (1 + x + x^2 + x^3+x*O(x^n))^(2*n+1)/(2*n+1), n)} /* Paul D. Hanna */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x*A+x*O(x^n))^m/m*sum(j=0, m, binomial(m, j)^2*x^j*A^(2*j))))); polcoeff(A, n, x)} /* Paul D. Hanna */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m*(1-x*A^2)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*A^(2*j))))); polcoeff(A, n, x)} /* Paul D. Hanna */

Crossrefs

Cf. A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

Cf. A200731.

Cf. A199874, A200074, A200075, A200718, A200719, A215576.

Sequence in context: A370441 A200740 A177133 * A193115 A270489 A335819

Adjacent sequences: A186238 A186239 A186240 * A186242 A186243 A186244

Keyword

nonn,easy

Author

Vladimir Kruchinin, Feb 15 2011

Status

approved