A296230 - OEIS

%I #6 Jan 24 2018 18:43:36

%S 1,1,0,1,0,1,1,0,4,0,6,13,9,48,101,147,542,1244,2385,8158,19191,44960,

%T 145355,356921,953648,2971797,7728368,22395844,68642687,189610373,

%U 577526057,1770461983,5170947386,16264118299,50488278032,154687144811,498055705248,1577949582705,5029555661992,16520308729413,53633742931559,176588771399224

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

%H Paul D. Hanna, <a href="/A296230/b296230.txt">Table of n, a(n) for n = 0..200</a>

%e G.f.: A(x) = 1 + x + x^3 + x^5 + x^6 + 4*x^8 + 6*x^10 + 13*x^11 + 9*x^12 + 48*x^13 + 101*x^14 + 147*x^15 + 542*x^16 + 1244*x^17 + 2385*x^18 + 8158*x^19 + 19191*x^20 + ...

%e such that

%e 1 = 1/A(x) + x/(1-x)/A(x)^3 + x^2/(1-x)^3/A(x)^6 + x^3/(1-x)^6/A(x)^10 + x^4/(1-x)^10/A(x)^15 + x^5/(1-x)^15/A(x)^21 + x^6/(1-x)^21/A(x)^28 + x^7/(1-x)^28/A(x)^36 + ...

%e Compare to the trivial identity:

%e 1 = 1/(1+x) + x*(1+x)/(1+x)^3 + x^2*(1+x)^3/(1+x)^6 + x^3*(1+x)^6/(1+x)^10 + x^4*(1+x)^10/(1+x)^15 + x^5*(1+x)^15/(1+x)^21 + ...

%o (PARI) {a(n) = my(A=[1],V); for(i=0,n, A = concat(A,0); V = Vec(sum(n=0,#A,1/(1-x +x*O(x^#A))^(n*(n+1)/2)*x^n/Ser(A)^((n+1)*(n+2)/2)) ); A[#A]=V[#A] ); A[n+1]}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A296231.

%K nonn

%O 0,9

%A _Paul D. Hanna_, Jan 24 2018