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A365516
Expansion of g.f. A(x) satisfying A(x) = (1 + 2*x*A(x))^2 / (1 + 2*x*A(x) - 2*x^3*A(x)^3).
2
1, 2, 4, 10, 32, 112, 400, 1464, 5520, 21296, 83456, 331136, 1328320, 5379200, 21959936, 90271904, 373347840, 1552438016, 6486311680, 27217331456, 114649525760, 484640538112, 2055185596416, 8740711936000, 37273693649920, 159340373710848, 682708771254272, 2931290431277056
OFFSET
0,2
COMMENTS
Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (A000108):
(1) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1.
(2) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1.
Related identity: [x^(n-1)] (1 + (n+1)*x*B(x))^n / B(x)^n = n*(n-1)^(n-1) for n >= 1 when B(x) = 1/(1 - 2*x).
Related identity: F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1), which holds formally for all Maclaurin series F(x). - Paul D. Hanna, Oct 03 2023
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = (1 + 2*x*A(x))^2 / (1 + 2*x*A(x) - 2*x^3*A(x)^3).
(2) A(x/B(x)) = B(x) where B(x) = (1 + 2*x)^2 / (1 + 2*x - 2*x^3).
(3) A(x) = (1/x) * Series_Reversion( x*(1 + 2*x - 2*x^3)/(1 + 2*x)^2 ).
(4) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+1)^(n-2) for n > 1.
(5) [x^(n-1)] (1 + (n-1)*x*A(x))^n / A(x)^n = -n*(n-3)^(n-2) for n > 1.
(6) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = (n^(n-1) - n*(n-2)^(n-2))/2 for n >= 1.
(7) A(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1). - Paul D. Hanna, Oct 03 2023
EXAMPLE
G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 32*x^4 + 112*x^5 + 400*x^6 + 1464*x^7 + 5520*x^8 + 21296*x^9 + 83456*x^10 + ...
where A(x) = (1 + 2*x*A(x))^2/(1 + 2*x*A(x) - 2*x^3*A(x)^3).
RELATED SERIES.
B(x) = A(x/B(x)) where B(x) = (1 + 2*x)^2/(1 + 2*x - 2*x^3) begins
B(x) = 1 + 2*x + 2*x^3 + 4*x^6 - 8*x^7 + 16*x^8 - 24*x^9 + 32*x^10 - 32*x^11 + 16*x^12 + 32*x^13 - 128*x^14 + ...
RELATED TABLE.
The table of coefficients of x^k in (1 + (n+1)*x*A(x))^n/A(x)^n begins:
n=1: [1, 0, 0, -2, -8, -24, -76, -272, ...];
n=2: [1, 2, 1, -4, -20, -64, -196, -664, ...];
n=3: [1, 6, 12, 2, -48, -192, -600, -1896, ...];
n=4: [1, 12, 54, 100, -23, -600, -2224, -6944, ...];
n=5: [1, 20, 160, 630, 1080, -696, -8660, -30960, ...];
n=6: [1, 30, 375, 2488, 9027, 14406, -15371, -144852, ...];
n=7: [1, 42, 756, 7546, 44800, 153552, 229376, -342728, ...];
n=8: [1, 56, 1372, 19192, 167222, 921400, 3026332, 4251528, ...]; ...
in which the main diagonal equals n*(n+1)^(n-2) for n > 1.
MATHEMATICA
a[n_] := Module[{A = {1}, m, x}, Do[AppendTo[A, 0]; m = Length[A]; A[[-1]] = Coefficient[(1 + (m + 1)*x*SeriesData[x, 0, A, 0, m + 1, 1])^m/SeriesData[x, 0, A, 0, m + 1, 1]^m, x, m - 1]/m - (m + 1)^(m - 2), {i, 1, n}]; A[[n + 1]]]; Table[a[n], {n, 0, 27}] (* Robert P. P. McKone, Sep 07 2023 *)
PROG
(PARI) /* Using series reversion */
{a(n) = my(A = (1/x)*serreverse(x*(1 + 2*x - 2*x^3)/(1 + 2*x +x*O(x^n))^2));
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Formula [x^(n-1)] (1 + (n+1)*x*A(x))^n/A(x)^n = n*(n+1)^(n-2) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m+1)^(m-2) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2023
STATUS
approved