Revision History for A185753
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing all changes.
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#9 by Paul D. Hanna at Thu Apr 26 21:24:05 EDT 2018
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#8 by Paul D. Hanna at Thu Apr 26 21:24:03 EDT 2018
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| FORMULA
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G.f. A(x) satisfies:
(4) [x^n] (x*(A(x) - 1)^2)' / (1 + x*(A(x) - 1)^2)^(n+1) = 0 for n>=1, where [x^n] F(x) denotes the coefficient of x^n in F(x). - Paul D. Hanna, Apr 26 2018
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| STATUS
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approved
editing
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#7 by Paul D. Hanna at Thu Apr 26 21:16:41 EDT 2018
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#6 by Paul D. Hanna at Thu Apr 26 21:16:38 EDT 2018
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| NAME
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G.f. satisfies: A(x/A(x)) = 1 + sqrt(x - x/A(x)).
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| FORMULA
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Let G(x) be the g.f. of A185754, then g.f. A(x) satisfies:
* (1) x + (A(x)-) - 1)^2 = G(x),
* (2) x* * A(( G(x)) = ) ) = G(x),
* G(3) G( x/A(x)) = ) ) = x.
G.f. A(x) satisfies:
(4) [x^n] (x*(A(x) - 1)^2)' / (1 + x*(A(x) - 1)^2)^(n+1) = 0 for n>=1, where [x^n] F(x) denotes the coefficient of x^n in F(x). - Paul D. Hanna, Apr 26 2018
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| EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 102*x^5 + 861*x^6 +...
RELATED SERIES.
A(x/A(x)) = 1 + x + x^3 + 5*x^4 + 37*x^5 + 329*x^6 + 3415*x^7 + 40328*x^8 + 532749*x^9 + 7777531*x^10 + 124315519*x^11 + ...
x - x/A(x) = x^2 + 2*x^4 + 10*x^5 + 75*x^6 + 668*x^7 + 6929*x^8 + 81684*x^9 + 1076987*x^10 + 15694214*x^11 + 250460767*x^12 + ...
sqrt(x - x/A(x)) = x + x^3 + 5*x^4 + 37*x^5 + 329*x^6 + 3415*x^7 + 40328*x^8 + ...
(A(x)-) - 1)^2 = x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 243*x^6 + 2016*x^7 +...
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| PROG
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((PARI) {a(n)=) = local(A=x+x^2); ); for(i=1, n, , A= = 2*A- - x-( -(x/serreverse(A+ + x^2*O(x^n))-)) - 1)^2); ); polcoeff(x/serreverse(A+ + x^2*O(x^n)), )), n)}
for(n=0, 30, print1(a(n), ", "))
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| CROSSREFS
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Cf. A185754.
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| STATUS
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approved
editing
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#5 by Russ Cox at Fri Mar 30 18:37:26 EDT 2012
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| AUTHOR
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Feb 02 2011
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Discussion
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| Fri Mar 30
| 18:37
| OEIS Server: https://oeis.org/edit/global/213
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#4 by T. D. Noe at Wed Feb 02 21:27:32 EST 2011
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#3 by Paul D. Hanna at Wed Feb 02 15:16:03 EST 2011
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| NAME
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allocated for Paul D. Hanna
G.f. satisfies: A(x/A(x)) = 1 + sqrt(x - x/A(x)).
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| DATA
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1, 1, 1, 3, 15, 102, 861, 8593, 98453, 1269924, 18187062, 286183564, 4907331899, 91082993194, 1819518069135, 38929958186607, 888318740697313, 21536467340324252, 552893064959418966, 14985039828839650746
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| OFFSET
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0,4
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| FORMULA
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Let G(x) be the g.f. of A185754, then g.f. A(x) satisfies:
* x + (A(x)-1)^2 = G(x),
* x*A(G(x)) = G(x),
* G(x/A(x)) = x.
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| EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 102*x^5 + 861*x^6 +...
The g.f. of A185754 begins:
G(x) = x + x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 243*x^6 + 2016*x^7 +...
where
(A(x)-1)^2 = x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 243*x^6 + 2016*x^7 +...
A(G(x)) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 243*x^5 + 2016*x^6 +...
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| PROG
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(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=2*A-x-(x/serreverse(A+x^2*O(x^n))-1)^2); polcoeff(x/serreverse(A+x^2*O(x^n)), n)}
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| CROSSREFS
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Cf. A185754.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Feb 02 2011
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| STATUS
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approved
proposed
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#2 by Paul D. Hanna at Wed Feb 02 15:13:08 EST 2011
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| KEYWORD
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allocating
allocated
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#1 by Paul D. Hanna at Wed Feb 02 15:13:08 EST 2011
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| NAME
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allocated for Paul D. Hanna
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| KEYWORD
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allocating
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| STATUS
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approved
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