A340895 - OEIS

%I #15 Mar 08 2021 12:08:25

%S 1,1,4,30,266,2578,26501,284016,3139627,35546887,410192986,4807352738,

%T 57070869569,684923802002,8296759274813,101314431713383,

%U 1245926701296828,15417454366803538,191837536701986616,2398860867832185041,30131118130388007583,379999736040117142018

%N G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^n)  =  Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(3*n+2)).

%C Equals row k = 3 of rectangular table A340940.

%F Given g.f. A(x), the following sums are all equal:

%F (1) B(x) = Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^n),

%F (2) B(x) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^(3*n+1)),

%F (3) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(3*n+2)),

%F (4) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+3)),

%F (5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+3*n) * (1 - x^2*A(x)^(2*n+3)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+3))),

%F (6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(3*n^2+3*n) * (1 - x^2*A(x)^(6*n+3)) / ((1 - x*A(x)^(3*n+1))*(1 - x*A(x)^(3*n+2)));

%F see the example section for the value of B(x).

%e G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 266*x^4 + 2578*x^5 + 26501*x^6 + 284016*x^7 + 3139627*x^8 + 35546887*x^9 + 410192986*x^10 + 4807352738*x^11 + ...

%e such that

%e B(x) = 1/(1-x) + x*A(x)^3/(1 - x*A(x)) + x^2*A(x)^6/(1 - x*A(x)^2) + x^3*A(x)^9/(1 - x*A(x)^3) + x^4*A(x)^12/(1 - x*A(x)^4) + ...

%e and

%e B(x) = 1/(1-x*A(x)) + x*A(x)^2/(1 - x*A(x)^4) + x^2*A(x)^4/(1 - x*A(x)^7) + x^3*A(x)^6/(1 - x*A(x)^10) + x^4*A(x)^8/(1 - x*A(x)^13) + ...

%e also

%e B(x) = 1/(1-x*A(x)^2) + x*A(x)/(1 - x*A(x)^5) + x^2*A(x)^2/(1 - x*A(x)^8) + x^3*A(x)^3/(1 - x*A(x)^11) + x^4*A(x)^4/(1 - x*A(x)^14) + ...

%e further

%e B(x) = 1/(1-x*A(x)^3) + x/(1 - x*A(x)^4) + x^2/(1 - x*A(x)^5) + x^3/(1 - x*A(x)^6) + x^4/(1 - x*A(x)^7) + x^5/(1 - x*A(x)^8) + ...

%e where

%e B(x) = 1 + 2*x + 6*x^2 + 29*x^3 + 203*x^4 + 1738*x^5 + 16574*x^6 + 168779*x^7 + 1797190*x^8 + 19770290*x^9 + 222969428*x^10 + 2564646302*x^11 + ...

%o (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0); H=A; A=concat(A,0);

%o H[#A-1] = -polcoeff( sum(m=0,#A, x^m/(1 - x*Ser(A)^(m+3)) ) - sum(m=0,#A, x^m*Ser(A)^m/(1 - x*Ser(A)^(3*m+2)) ), #A)/2; A=H); A[n+1] }

%o for(n=0,30,print1(a(n),", "))

%Y Cf. Cf. A340940, A340941, A340942, A340943, A340894, A341376.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 26 2021