A264233 G.f. satisfies: A(x)^2 = A( x^2/(1-12*x)^2 ).

1, 12, 150, 1944, 25977, 355932, 4975974, 70684920, 1016911392, 14778827136, 216547264296, 3194332332192, 47384274750705, 706221689838300, 10568432343600990, 158713925474269080, 2390963478663939555, 36119150645827725540, 547001314170524048970, 8302813348383238118760, 126288497159001902128185, 1924561894757711270308380

Offset

1,2

Comments

Radius of convergence is r = 1/16 where r = r^2/(1-12*r)^2 with A(r) = 1.

Compare to: C(x)^2 = C( x^2/(1-2*x)^2 ) where C(x) = (1-2*x-sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers A000108.

Links

Table of n, a(n) for n=1..22.

Formula

G.f. satisfies:

(1) A(x) = -A( -x/(1-24*x) ).

(2) A(x^2) = A( x/(1+12*x) )^2 = A( -x/(1-12*x) )^2.

(3) A( x/(1+6*x)^2 ) = -A( -x/(1-6*x)^2 ), an odd function.

(4) A( x/(1+6*x)^2 )^2 = A( x^2/(1+36*x^2)^2 ), an even function.

(5) A( x/(1+9*x) ) = G(x) = Sum_{n>=1} A264225(n)*x^n where G(x)^2 = G( x^2/(1-6*x) ).

(6) A( x/(1+15*x) ) = -G(-x) = Sum_{n>=1} (-1)^(n-1) * A264225(n)*x^n where G(x)^2 = G( x^2/(1-6*x) ).

Sum_{k=0..n} binomial(n,k) *(-12)^(n-k) * a(k+1) = 0 for odd n.

Sum_{k=0..n} binomial(n,k) * (-9)^(n-k) * a(k+1) = A264225(n+1) for n>=0.

Sum_{k=0..n} binomial(n,k) *(-15)^(n-k) * a(k+1) = (-1)^n * A264225(n+1) for n>=0.

Example

G.f.: A(x) = x + 12*x^2 + 150*x^3 + 1944*x^4 + 25977*x^5 + 355932*x^6 + 4975974*x^7 + 70684920*x^8 + 1016911392*x^9 + 14778827136*x^10 + 216547264296*x^11 +...
where A( x^2/(1-12*x)^2 ) = A(x)^2,
A( x^2/(1-12*x)^2 ) = x^2 + 24*x^3 + 444*x^4 + 7488*x^5 + 121110*x^6 + 1918512*x^7 + 30066552*x^8 + 468571392*x^9 + 7281721209*x^10 + 113007681720*x^11 +...
Also, A( x/(1+12*x) ) = A(x^2)^(1/2),
A( x/(1+12*x) ) = x + 6*x^3 + 57*x^5 + 630*x^7 + 7584*x^9 + 96552*x^11 + 1277937*x^13 + 17393454*x^15 + 241666275*x^17 + 3410638362*x^19 + 48723929721*x^21 +...
Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then
B(x) = 1 + 12*x + 6*x^2 - 15*x^4 + 90*x^6 - 660*x^8 + 5310*x^10 - 45765*x^12 + 413640*x^14 - 3864345*x^16 + 37014120*x^18 - 361577790*x^20 +...+ A264413(n)*x^(2*n) +...
such that B(x) = F(x^2) + 12*x = F(x)^2 where F(x) is the g.f. of A264413.

Prog

(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-12*x +x*O(x^n))^2) ) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A264413, A264225, A264232.

Sequence in context: A305544 A056351 A056345 * A068768 A053507 A060917

Adjacent sequences: A264230 A264231 A264232 * A264234 A264235 A264236

Keyword

nonn

Author

Paul D. Hanna, Nov 17 2015

Status

approved