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A338306
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O.g.f. A(x) satisfies: A(x) = 1/(1 - x * (d/dx x + x^2*A(x)) ).
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1
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1, 1, 3, 8, 29, 106, 438, 1869, 8499, 39926, 195332, 983262, 5099640, 27104535, 147565543, 820663322, 4658281691, 26943982072, 158681719362, 950517101448, 5787148086216, 35785974189746, 224625453101666, 1430389538402366, 9236235576715260, 60448170602414931, 400818049588802391
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OFFSET
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0,3
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LINKS
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FORMULA
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O.g.f. A(x) satisfies:
(1) A(x) = 1/(1 - x * (d/dx x + x^2*A(x)) ).
(2) 0 = [x^n] exp( n * (x + x^2*A(x)) ) / A(x), for n > 0.
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 29*x^4 + 106*x^5 + 438*x^6 + 1869*x^7 + 8499*x^8 + 39926*x^9 + 195332*x^10 + 983262*x^11 + 5099640*x^12 + ...
where
1/A(x) = 1 - x - 2*x^2 - 3*x^3 - 12*x^4 - 40*x^5 - 174*x^6 - 742*x^7 - 3504*x^8 - 16821*x^9 - 84990*x^10 - ...
and
d/dx x + x^2*A(x) = 1 + 2*x + 3*x^2 + 12*x^3 + 40*x^4 + 174*x^5 + 742*x^6 + 3504*x^7 + 16821*x^8 + + 84990*x^9 + ...
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PROG
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(PARI) /* A(x) = 1/(1 - x*( d/dx x + x^2*A(x) )) */
{a(n) = my(A=1+x); for(i=1, n, A=1/(1 - x*deriv(x + x^2*A+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* [x^n] exp( n * (x + x^2*A(x)') ) / A(x) = 0 */
{a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*(x+x^2*Ser(A)) ) / Ser(A) )[m] ); A[n+1]}
for(n=0, 20, 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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