Revision History for A291314
(Underlined text is an addition;
strikethrough text is a deletion.)
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#7 by Vaclav Kotesovec at Mon Aug 28 07:15:09 EDT 2017
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#6 by Vaclav Kotesovec at Mon Aug 28 07:15:04 EDT 2017
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| FORMULA
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a(n) ~ c * d^n / n^(3/2), where d = 10.18955737678412420186395666504... and c = 0.037604336950996597... - Vaclav Kotesovec, Aug 28 2017
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approved
editing
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#5 by Paul D. Hanna at Tue Aug 22 12:51:34 EDT 2017
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#4 by Paul D. Hanna at Tue Aug 22 12:51:30 EDT 2017
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| LINKS
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Paul D. Hanna, <a href="/A291314/b291314.txt">Table of n, a(n) for n = 1..300</a>
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approved
editing
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#3 by Paul D. Hanna at Tue Aug 22 12:49:00 EDT 2017
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#2 by Paul D. Hanna at Tue Aug 22 12:48:57 EDT 2017
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| NAME
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allocated for Paul D. Hanna
G.f. A(x) satisfies: A( 2*A(x)^2 - 8*A(x)^3 ) = 2*x^2.
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| DATA
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1, 2, 8, 56, 400, 3072, 24544, 203520, 1728256, 14967296, 131689472, 1173936128, 10579907072, 96238768128, 882437177344, 8147574407168, 75685465759744, 706854135595008, 6633217371029504, 62514337980088320, 591441701724880896, 5615172282703937536, 53480608406362914816, 510849109679635693568, 4892689722718505271296
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| OFFSET
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1,2
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| FORMULA
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G.f. A(x) satisfies: A( sqrt( A(2*x^2 - 8*x^3)/2 ) ) = x.
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| EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 8*x^3 + 56*x^4 + 400*x^5 + 3072*x^6 + 24544*x^7 + 203520*x^8 + 1728256*x^9 + 14967296*x^10 + 131689472*x^11 + 1173936128*x^12 + 10579907072*x^13 + 96238768128*x^14 + 882437177344*x^15 + 8147574407168*x^16 +...
such that A( 2*A(x)^2 - 8*A(x)^3 ) = 2*x^2.
RELATED SERIES.
2*A(x)^2 - 8*A(x)^3 = 2*x^2 - 8*x^4 - 256*x^8 + 512*x^10 - 10240*x^12 + 69632*x^14 - 835584*x^16 + 7929856*x^18 - 81002496*x^20 + 791674880*x^22 - 8468299776*x^24 + 84863352832*x^26 - 913569742848*x^28 + 9452686147584*x^30 +...
Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:
Ai(x) = x - 2*x^2 - 16*x^4 + 16*x^5 - 160*x^6 + 544*x^7 - 3264*x^8 + 15488*x^9 - 79104*x^10 + 386560*x^11 - 2067456*x^12 + 10359296*x^13 - 55759872*x^14 + 288473088*x^15 - 1546903552*x^16 + 8224194560*x^17 - 44310626304*x^18 + 238776467456*x^19 - 1295524724736*x^20 +...
where Ai(x) = sqrt( A(2*x^2 - 8*x^3)/2 )
and Ai( 2*Ai(x)^2 ) = 2*x^2 - 8*x^3.
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| PROG
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(PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = x*Ser(V); V[#V] = -polcoeff(subst(G=A, x, 2*A^2 - 8*A^3 ), #V+1)/4); V[n]}
for(n=1, 30, print1(a(n), ", "))
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| CROSSREFS
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Cf. A271961, A291313, A291315.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna, Aug 22 2017
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| STATUS
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approved
editing
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#1 by Paul D. Hanna at Mon Aug 21 22:41:56 EDT 2017
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| NAME
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allocated for Paul D. Hanna
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| KEYWORD
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allocated
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| STATUS
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approved
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