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A192947
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G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^4 - 1)^n.
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5
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1, 1, 4, 38, 444, 5805, 81284, 1192144, 18078660, 281172017, 4460264072, 71886775636, 1173832034804, 19377733213699, 322866234066016, 5422493523853024, 91701823351874276, 1560232214582865621, 26688686144512908492
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OFFSET
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0,3
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COMMENTS
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Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.
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LINKS
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FORMULA
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G.f. A(x) equals the formal inverse of function (x-1)*(2-x^4).
G.f. satisfies: A(x) = 1 + x/(2 - A(x)^4).
G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^4 ).
Recurrence: 2048*(n-3)*(n-2)*(n-1)*n*(150965*n^3 - 1545075*n^2 + 5236171*n - 5868843)*a(n) = 256*(n-3)*(n-2)*(n-1)*(21135100*n^4 - 248013150*n^3 + 1057496990*n^2 - 1920941631*n + 1231831665)*a(n-1) + 32*(n-3)*(n-2)*(204104680*n^5 - 2905360120*n^4 + 16233323207*n^3 - 44427578447*n^2 + 59458979970*n - 31076616060)*a(n-2) + 8*(n-3)*(372581620*n^6 - 6607607250*n^5 + 48312584548*n^4 - 186256666080*n^3 + 398929297639*n^2 - 449551482162*n + 207932551470)*a(n-3) + 5*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(150965*n^3 - 1092180*n^2 + 2598916*n - 2026782)*a(n-4). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 35/8 + 1/(8*sqrt(3/(4351 + (6495403 + 452895*sqrt(831))^(1/3)/2^(2/3) - 40027/(12990806 + 905790*sqrt(831))^(1/3)))) + 1/2*sqrt(4351/24 - (6495403 + 452895*sqrt(831))^(1/3)/(48*2^(2/3)) + 40027/(48*(12990806 + 905790*sqrt(831))^(1/3)) + 27661/4*sqrt(3/(4351 + (6495403 + 452895*sqrt(831))^(1/3)/2^(2/3) - 40027/(12990806 + 905790*sqrt(831))^(1/3)))) = 18.6600216048327281... is the root of the equation -3125 - 19744*d - 43264*d^2 - 35840*d^3 + 2048*d^4 = 0 and c = 0.047032341973499367520535028629... - Vaclav Kotesovec, Sep 17 2013
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 38*x^3 + 444*x^4 + 5805*x^5 +...
where (A(x) - 1)*(2 - A(x)^4) = x
and A(x - 4*x^2 - 6*x^3 - 4*x^4 - x^5) = 1 + x.
Related expansions.
(A(x)^4-1) = 4*x + 22*x^2 + 204*x^3 + 2377*x^4 + 31036*x^5 +...
(A(x)^4-1)^2 = 16*x^2 + 176*x^3 + 2116*x^4 + 27992*x^5 +...
(A(x)^4-1)^3 = 64*x^3 + 1056*x^4 + 15600*x^5 + 232456*x^6 +...
(A(x)^4-1)^4 = 256*x^4 + 5632*x^5 + 98688*x^6 + 1640576*x^7 +...
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MATHEMATICA
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CoefficientList[1+InverseSeries[Series[2*x-x*(1+x)^4, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Sep 17 2013 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^4-1+x*O(x^n))^m)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+serreverse(2*x-x*(1+x)^4+x^2*O(x^n))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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