A234313
E.g.f. satisfies: A'(x) = A(x)^5 * A(-x) with A(0) = 1.
Original entry on oeis.org
1, 1, 4, 34, 376, 5896, 107104, 2445664, 61835776, 1853785216, 60075541504, 2229983878144, 88157067006976, 3901637972801536, 182049480718741504, 9356335870657921024, 503257631887961522176, 29455739077723718189056, 1794347026494847887867904, 117825990265521485020463104
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 376*x^4/4! + 5896*x^5/5! +...
Related series.
A(x)^5 = 1 + 5*x + 40*x^2/2! + 470*x^3/3! + 7120*x^4/4! + 134000*x^5/5! +...
A(x)^3 = 1 + 3*x + 18*x^2/2! + 180*x^3/3! + 2376*x^4/4! + 40608*x^5/5! +...
Note that 1 - 1/A(x)^3 is an odd function:
1 - 1/A(x)^3 = 3*x + 18*x^3/3! + 1728*x^5/5! + 496368*x^7/7! + 287929728*x^9/9! +...
where Series_Reversion((1 - 1/A(x)^3)/3) = Integral (1-9*x^2)^(1/3) dx.
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CoefficientList[1/(1 - 3*InverseSeries[Series[Integrate[(1-9*x^2)^(1/3),x],{x,0,20}],x])^(1/3),x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2014 *)
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{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^5*subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=local(A=1); A=1/(1-3*serreverse(intformal((1-9*x^2 +x*O(x^n))^(1/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
A235329
E.g.f. satisfies: A'(x) = A(x)^3 / A(-x) with A(0) = 1.
Original entry on oeis.org
1, 1, 4, 22, 184, 1888, 24352, 364336, 6372352, 125098624, 2765195776, 67161837568, 1795080211456, 51946830487552, 1628857441189888, 54705106541123584, 1968709261466042368, 75262309701303402496, 3057220808668673081344, 131069643224297960046592
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 22*x^3/3! + 184*x^4/4! + 1888*x^5/5! +...
Related series.
A(x)^3 = 1 + 3*x + 18*x^2/2! + 144*x^3/3! + 1512*x^4/4! + 19224*x^5/5! +...
Note that 1 - 1/A(x)^3 is an odd function:
1 - 1/A(x)^3 = 3*x - 18*x^3/3! - 216*x^5/5! - 18144*x^7/7! - 3483648*x^9/9! +...
where Series_Reversion((1 - 1/A(x)^3)/3) = Integral 1/(1-9*x^2)^(1/3) dx.
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CoefficientList[1/(1 - 3*InverseSeries[Series[Integrate[1/(1-9*x^2)^(1/3),x],{x,0,20}],x])^(1/3),x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2014 *)
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/* By definition A'(x) = A(x)^3 / A(-x): */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^3/subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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/* E.g.f. 1/(1 - 3*Series_Reversion(Integral (1-9*x)^(1/3) dx))^(1/3): */
{a(n)=local(A=1);A=1/(1-3*serreverse(intformal(1/(1-9*x^2 +x*O(x^n))^(1/3))))^(1/3);n!*polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))
A235321
E.g.f. satisfies: A'(x) = A(x)^4 * A(-x)^2 with A(0) = 1.
Original entry on oeis.org
1, 1, 2, 10, 56, 496, 4592, 58240, 753536, 12428416, 206700032, 4194910720, 85291222016, 2054623707136, 49390297628672, 1376983620812800, 38210541166493696, 1209980100295622656, 38070418025153626112, 1349585751030491054080, 47479274652991633227776
Offset: 0
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 10*x^3/3! + 56*x^4/4! + 496*x^5/5! +...
Related series.
A(x)^2 = 1 + 2*x + 6*x^2/2! + 32*x^3/3! + 216*x^4/4! + 1952*x^5/5! +...
A(x)^4 = 1 + 4*x + 20*x^2/2! + 136*x^3/3! + 1160*x^4/4! + 12064*x^5/5! +...
A(x)^5 = 1 + 5*x + 30*x^2/2! + 230*x^3/3! + 2160*x^4/4! + 24200*x^5/5! +...
where 3 - 15*A(x) + 20*A(x)^2 = (8 - 15*x)*A(x)^5.
Note that 1 - 1/A(x) is an odd function that begins:
1 - 1/A(x) = x + 4*x^3/3! + 136*x^5/5! + 12544*x^7/7! + 2240896*x^9/9! +...
where Series_Reversion(1 - 1/A(x)) = x - 2*x^3/3 + x^5/5.
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CoefficientList[1/(1 - InverseSeries[Series[x - 2*x^3/3 + x^5/5, {x, 0, 20}], x]),x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2014 *)
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/* By definition, A'(x) = A(x)^4 * A(-x)^2: */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^4*subst(A,x,-x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))
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/* By equation: 3 - 15*A(x) + 20*A(x)^2 - (8 - 15*x)*A(x)^5 = 0 */
{a(n)=local(A=1);for(i=0,n,A=(3+20*A^2-(8-15*x)*A^5)/15 +x*O(x^n));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
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/* E.g.f. 1/(1 - Series_Reversion(Integral (1-x^2)^2 dx)): */
{a(n)=local(A=1); A=1/(1-serreverse(x-2*x^3/3+x^5/5 +x^2*O(x^n)));n!*polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))
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/* E.g.f. 1 + Series_Reversion(Integral (1+2*x)^2/(1+x)^6 dx): */
{a(n)=local(A=1); A=1+serreverse( intformal((1+2*x)^2/(1+x +x^2*O(x^n))^6 ));n!*polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))
Showing 1-3 of 3 results.