A341954 G.f. A(x) satisfies: A(x) = 1/((1-x*B(x)^2)*(1-x*C(x)^3)) such that B(x) = 1/((1-x*A(x))*(1-x*C(x)^3)) and C(x) = 1/((1-x*A(x))*(1-x*B(x)^2)) are the g.f.s of A341955 and A341956, respectively.

1, 2, 13, 99, 839, 7606, 72190, 708294, 7126305, 73125017, 762337935, 8051642336, 85971106450, 926481778388, 10064065073450, 110080177918855, 1211363817278035, 13401851361051323, 148978925959605763, 1663181275248666597

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

G.f. A(x) and related series B(x) and C(x) satisfy:

(1a) A(x) = 1/((1 - x*B(x)^2)*(1 - x*C(x)^3)),

(1b) B(x) = 1/((1 - x*A(x))*(1 - x*C(x)^3)),

(1c) C(x) = 1/((1 - x*A(x))*(1 - x*B(x)^2)).

(2a) A(x) = B(x)*C(x) * (1 - x*A(x))^2,

(2b) B(x) = A(x)*C(x) * (1 - x*B(x)^2)^2,

(2c) C(x) = A(x)*B(x) * (1 - x*C(x)^3)^2.

(3a) A(x)/(1 - x*A(x)) = B(x)/(1 - x*B(x)^2) = C(x)/(1 - x*C(x)^3) = sqrt(A(x)*B(x)*C(x)),

(3b) A(x)*B(x)*C(x) = 1/((1-x*A(x))^2*(1-x*B(x)^2)^2*(1-x*C(x)^3)^2).

Example

G.f.: A(x) = 1 + 2*x + 13*x^2 + 99*x^3 + 839*x^4 + 7606*x^5 + 72190*x^6 + 708294*x^7 + 7126305*x^8 + 73125017*x^9 + 762337935*x^10 + ...
such that A(x) = 1/((1-x*B(x)^2)*(1-x*C(x)^3)) where
B(x) = 1 + 2*x + 11*x^2 + 80*x^3 + 659*x^4 + 5865*x^5 + 54954*x^6 + 534087*x^7 + 5334509*x^8 + 54423368*x^9 + 564713959*x^10 + ...
C(x) = 1 + 2*x + 9*x^2 + 61*x^3 + 489*x^4 + 4283*x^5 + 39702*x^6 + 382899*x^7 + 3802403*x^8 + 38618535*x^9 + 399277260*x^10 + ...
RELATED SERIES.
A(x)*B(x)*C(x) = 1 + 6*x + 45*x^2 + 380*x^3 + 3438*x^4 + 32584*x^5 + 319358*x^6 + 3210482*x^7 + 32921947*x^8 + 343030506*x^9 + ...
sqrt(A(x)*B(x)*C(x)) = 1 + 3*x + 18*x^2 + 136*x^3 + 1149*x^4 + 10397*x^5 + 98558*x^6 + 966157*x^7 + 9714366*x^8 + 99631288*x^9 + ...
where
sqrt(A(x)*B(x)*C(x)) = A(x)/(1-x*A(x)) = B(x)/(1-x*B(x)^2) = C(x)/(1-x*C(x)^3).
A(x)^2 = 1 + 4*x + 30*x^2 + 250*x^3 + 2243*x^4 + 21142*x^5 + 206419*x^6 + 2069226*x^7 + 21172635*x^8 + 220227386*x^9 + ...
B(x)^2 = 1 + 4*x + 26*x^2 + 204*x^3 + 1759*x^4 + 16126*x^5 + 154266*x^6 + 1522460*x^7 + 15387035*x^8 + 158457396*x^9 + ...
C(x)^2 = 1 + 4*x + 22*x^2 + 158*x^3 + 1303*x^4 + 11620*x^5 + 109059*x^6 + 1061358*x^7 + 10612685*x^8 + 108371282*x^9 + ...
C(x)^3 = 1 + 6*x + 39*x^2 + 299*x^3 + 2550*x^4 + 23229*x^5 + 221256*x^6 + 2176734*x^7 + 21947076*x^8 + 225589243*x^9 + ...

Prog

(PARI) {a(n) = my(A=1, B=1, C=1); for(i=1, n,
A = 1/((1-x*B^2)*(1-x*C^3) +x*O(x^n));
B = 1/((1-x*A^1)*(1-x*C^3) +x*O(x^n));
C = 1/((1-x*A^1)*(1-x*B^2) +x*O(x^n)); );
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A341955 (B(x)), A341956 (C(x)).

Sequence in context: A064325 A365155 A123619 * A187746 A030519 A141116

Adjacent sequences: A341951 A341952 A341953 * A341955 A341956 A341957

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2021

Status

approved