A299854 G.f. S(x) satisfies C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 72*x.

36, -144, 864, -6048, 46080, -370656, 3096576, -26604864, 233570304, -2086063200, 18893242368, -173111152320, 1601754365952, -14945262816960, 140461536706560, -1328495714939520, 12635295568625664, -120770748612067680, 1159474181591531520, -11176086736640548800, 108114032779214192640, -1049288769526156568640, 10214201953998140866560, -99701511082612622322048

Offset

2,1

Comments

a(n) = -(-1)^n * 36 * A214377(n-2) for n>=2.

Links

Table of n, a(n) for n=2..25.

Formula

The functions C = C(x) and S = S(x) satisfy:

(1a) sqrt(C) - sqrt(S) = 1.

(1b) C'*sqrt(S) = S'*sqrt(C) = 72*x.

(1c) C' = 72*x/sqrt(S).

(1d) S' = 72*x/sqrt(C).

Integrals.

(2a) C = 1 + Integral 72*x/sqrt(S) dx.

(2b) S = Integral 72*x/sqrt(C) dx.

(2c) C = 1 + Integral S'*sqrt(C/S) dx.

(2d) S = Integral C'*sqrt(S/C) dx.

Exponentials.

(3a) sqrt(C) = exp( Integral 36*x/(C*sqrt(S)) dx ).

(3b) sqrt(S) = 6*x*exp( Integral 36*x/(S*sqrt(C)) - 1/x dx ).

(3c) C - S = exp( Integral 72*x/(C*sqrt(S) + S*sqrt(C)) dx ).

(3d) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).

Functional equations.

(4a) C = 1/3 - 36*x^2 + (2/3)*C^(3/2).

(4b) S = 36*x^2 - (2/3)*S^(3/2).

Explicit solutions.

(5a) C(x) = 1 + Sum_{n>=1} 2*(-4)^n*binomial(3*n/2,n)/((3*n-2)*(3*n-4)) * x^n.

(5b) S(x) = 36*x^2 + Sum_{n>=3} 18*(-4)^n*(3*n-3)*binomial(3*n/2-2,n)/((3*n-4)*(3*n-6)) * x^n.

(5c) sqrt(C(x)) = 1 + Sum_{n>=1} -(-4)^n * binomial(3*n/2,n)/(3*n-2) * x^n.

Formulas for terms.

a(n) = 18*(-4)^n * (3*n-3) * binomial(3*n/2-2,n) / ((3*n-4)*(3*n-6)) for n>=3, with a(2) = 36.

Example

G.f.: S(x) = 36*x^2 - 144*x^3 + 864*x^4 - 6048*x^5 + 46080*x^6 - 370656*x^7 + 3096576*x^8 - 26604864*x^9 + 233570304*x^10 + ...
RELATED SERIES.
C(x) = 1 + 12*x + 12*x^2 - 24*x^3 + 96*x^4 - 504*x^5 + 3072*x^6 - 20592*x^7 + 147456*x^8 - 1108536*x^9 + 8650752*x^10 + ...
C(x)^(1/2) = 1 + 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
sqrt(S(x)) = 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
where C(x)^(1/2) - S(x)^(1/2) = 1
and C'*sqrt(S) = S'*sqrt(C) = 72*x.

Prog

(PARI) {a(n) = my(C=1, S=x^2); for(i=0, n, C = 1 + intformal( 72*x/sqrt(S +x^3*O(x^n)) ); S = intformal( 72*x/sqrt(C) ) ); polcoeff(S, n)}
for(n=2, 30, print1(a(n), ", "))
(PARI) {a(n) = if(n<2, 0, n==2, 36, 18*(-4)^n * (3*n-3) * binomial(3*n/2-2, n) / ((3*n-4)*(3*n-6)) )}
for(n=2, 30, print1(a(n), ", "))

Crossrefs

Cf. A299853, A299855, A214377.

Sequence in context: A288488 A262789 A110754 * A154711 A162992 A037072

Adjacent sequences: A299851 A299852 A299853 * A299855 A299856 A299857

Keyword

sign

Author

Paul D. Hanna, Feb 20 2018

Status

approved