A341954 - OEIS

%I #7 Mar 01 2021 11:31:02

%S 1,2,13,99,839,7606,72190,708294,7126305,73125017,762337935,

%T 8051642336,85971106450,926481778388,10064065073450,110080177918855,

%U 1211363817278035,13401851361051323,148978925959605763,1663181275248666597

%N 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.

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

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

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

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

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

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

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

%F (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)),

%F (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).

%e 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 + ...

%e such that A(x) = 1/((1-x*B(x)^2)*(1-x*C(x)^3)) where

%e 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 + ...

%e 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 + ...

%e RELATED SERIES.

%e 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 + ...

%e 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 + ...

%e where

%e 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).

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e 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 + ...

%o (PARI) {a(n) = my(A=1,B=1,C=1); for(i=1,n,

%o A = 1/((1-x*B^2)*(1-x*C^3) +x*O(x^n));

%o B = 1/((1-x*A^1)*(1-x*C^3) +x*O(x^n));

%o C = 1/((1-x*A^1)*(1-x*B^2) +x*O(x^n)););

%o polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 25 2021