A258657
E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)^2 * Integral 1/A(x)^6 dx dx ).
Original entry on oeis.org
1, 1, 3, 63, 1449, 79569, 4933467, 538446447, 64308790161, 11702872619361, 2287996769790963, 625623383080392543, 181555646645074399929, 69630517380932205676209, 28096571683140898776853707, 14391876463069615780007165967, 7705942233644896318623038389281, 5082509306223148713110677899062721
Offset: 0
E.g.f. A(x) = 1 + x^2/2! + 3*x^4/4! + 63*x^6/6! + 1449*x^8/8! + 79569*x^10/10! +...
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a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Sqrt @ JacobiDC[ x, -1], {x, 0, m}]]]; (* Michael Somos, Jun 17 2016 *)
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{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^2 * intformal(1/A^6 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))
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{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( 1/A^2 * intformal(A^6 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))
A258658
E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)^3 * Integral 1/A(x)^9 dx dx ).
Original entry on oeis.org
1, 1, 3, 123, 3129, 313929, 23062347, 4461062067, 655619300721, 207299066045841, 51139145307699603, 23947711418548452843, 8930007221716001596329, 5800583064218449362313689, 3049682921576823757255485147, 2622347479175212075411836474147, 1849810669560927151969244969258721
Offset: 0
E.g.f. A(x) = 1 + x^2/2! + 3*x^4/4! + 123*x^6/6! + 3129*x^8/8! + 313929*x^10/10! +...
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{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^3 * intformal(1/A^9 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))
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{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( 1/A^3 * intformal(A^9 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))
A261000
Unordered even-degree bilabeled increasing trees on 2n+1 nodes.
Original entry on oeis.org
1, 3, 189, 68607, 82908441, 251944606683, 1618221395188629, 19514714407120367127, 405452689572115086887601, 13596354857453497541480646963, 699110237190377161907394095173869, 52888313306236766686682435536884784047
Offset: 0
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A261000aer := proc(n)
option remember;
local a,nloc,j,k,l;
if n = 1 then
1;
else
nloc := n-2 ;
a :=0 ;
for j from 0 to nloc-1 do
for k from 0 to nloc-1-j do
l := nloc-1-j-k ;
if l >= 0 then
a := a+procname(j+1)*procname(k+1)*procname(l+1) * (2*nloc+1)!/(2*j+1)!/(2*k+1)!/(2*l+1)! ;
end if;
end do:
end do:
%/2 ;
end if;
end proc:
A261000 := proc(n)
A261000aer(2*n+1) ;
end proc:
seq(A261000(n),n=0..15) ; # R. J. Mathar, Aug 18 2015
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terms = 12; nmax = 4 terms; A = 1; Do[A = Exp[Integrate[A^(1/2)*Integrate[1/A^(3/2), x], x] + O[x]^nmax], nmax]; A258659 = CoefficientList[A, x^2]*Range[0, nmax - 2, 2]!;
a[n_] := A258659[[2 n + 1]];
Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Nov 27 2017 *)
a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[ JacobiSD[x, 1/2], {x, 0, 4*n+1}]]; (* Michael Somos, Sep 03 2022 *)
a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[ x*Sqrt[1/x^2 / WeierstrassP[x, {1, 0}]], {x, 0, 4*n+1}]]; (* Michael Somos, Jul 02 2024 *)
a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[
InverseSeries[ Series[ x * Hypergeometric2F1[1/4, 1/2, 5/4, x^4/4], {x, 0, 4*n+1}]], {x, 0, 4*n+1}]]; (* Michael Somos, Jul 02 2024 *)
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{a(n) = if( n<0, 0, my(m = 4*n + 1); m! * polcoeff( serreverse( intformal( 1 / sqrt(1 + x^4/4 + x * O(x^m)) ) ), m))}; /* Michael Somos, Jun 17 2017 */
Showing 1-3 of 3 results.
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