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

1, 2, 11, 80, 659, 5865, 54954, 534087, 5334509, 54423368, 564713959, 5941244370, 63230204938, 679510980507, 7363532850004, 80372780735971, 882818219523503, 9751004973855748, 108236495732967482, 1206750569591821120, 13507907804245679450

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f. B(x) and related series A(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.: 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 + ...
such that B(x) = 1/((1 - x*A(x))*(1 - x*C(x)^3)) where
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 + ...
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(B, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A341954 (A(x)), A341956 (C(x)).

Sequence in context: A320095 A099661 A027110 * A118969 A181068 A056846

Adjacent sequences: A341952 A341953 A341954 * A341956 A341957 A341958

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2021

Status

approved