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| NAME
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allocated for Paul D. Hanna
G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(3*n) + (-1)^n*A(x))^n.
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| DATA
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1, 1, 2, 3, 4, 6, 11, 19, 33, 60, 110, 201, 372, 696, 1305, 2456, 4654, 8855, 16896, 32366, 62217, 119910, 231704, 448879, 871531, 1695541, 3305057, 6454033, 12624041, 24731009, 48520396, 95324187, 187517959, 369329721, 728262059, 1437591201, 2840751293
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| OFFSET
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1,3
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| FORMULA
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G.f. A(x) satisfies:
(1) 1 - x = Sum_{n>=0} ( x^(3*n) + (-1)^n*A(x) )^n.
(2) 1 - x = Sum_{n>=0} x^(3*n^2) / (1 + (-1)^n*x^(3*n)*A(x))^(n+1).
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| EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 11*x^7 + 19*x^8 + 33*x^9 + 60*x^10 + 110*x^11 + 201*x^12 + 372*x^13 + 696*x^14 + ...
where
1 - x = 1 + (x^3 - A(x)) + (x^6 + A(x))^2 + (x^9 - A(x))^3 + (x^12 + A(x))^4 + (x^15 - A(x))^5 + (x^18 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^3/(1 - x^3*A(x))^2 + x^12/(1 + x^6*A(x))^3 + x^27/(1 - x^9*A(x))^4 + x^48/(1 + x^12*A(x))^5 + ...
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| PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (x^(3*m) + (-1)^m*x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A\3), x^(3*m^2)/(1 + (-x)^(3*m)*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
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| CROSSREFS
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Cf. A317997, A352818, A352820, A352821.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna, Apr 05 2022
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| STATUS
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approved
editing
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