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A275758
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G.f. satisfies: A(x) = x + A( A(x)^4 - A(x)^10 ).
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4
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1, 1, 4, 21, 126, 817, 5574, 39418, 286286, 2122491, 15995696, 122166551, 943430560, 7353998931, 57783603764, 457176705018, 3639000808140, 29119701312548, 234120338807316, 1890257713736568, 15319612051101438, 124583720191974904, 1016307862050772614, 8314217332992596050, 68193993494598345010, 560671685990956975367, 4619857060146629819160, 38144728242794104501561, 315546193363448088862064, 2614910268303053285326541
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OFFSET
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1,3
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COMMENTS
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Compare to: G(x) = x + G( G(x)^4 - G(x)^16 ) holds when G(x) = x + G(x)^4 is a g.f. of A002293.
Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x - A(x^4 - x^10)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^4-x^10)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^4-x^10)^(n+1)/x / (n+1)! ).
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EXAMPLE
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G.f.: A(x) = x + x^4 + 4*x^7 + 21*x^10 + 126*x^13 + 817*x^16 + 5574*x^19 + 39418*x^22 + 286286*x^25 + 2122491*x^28 + 15995696*x^31 + 122166551*x^34 +...
such that A(x) = x + A( A(x)^4 - A(x)^10 ).
RELATED SERIES.
A(x)^4 = x^4 + 4*x^7 + 22*x^10 + 136*x^13 + 901*x^16 + 6248*x^19 + 44758*x^22 + 328520*x^25 + 2457286*x^28 + 18659736*x^31 + 143455026*x^34 +...
A(x)^10 = x^10 + 10*x^13 + 85*x^16 + 690*x^19 + 5520*x^22 + 44002*x^25 + 351045*x^28 + 2808040*x^31 + 22537355*x^34 + 181530280*x^37 + 1467320874*x^40 +...
A(x^4 - x^10) = x^4 - x^10 + x^16 - 4*x^22 + 10*x^28 - 32*x^34 + 106*x^40 - 350*x^46 + 1211*x^52 - 4242*x^58 + 15083*x^64 - 54404*x^70 + 198114*x^76 +...
where Series_Reversion(A(x)) = x - A(x^4 - x^10).
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, 3*n, A = x + subst(A, x, A^4 - A^10 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
for(n=1, 30, print1(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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