A195981
Coefficients of expansion of 1/xi_0(y) (see A195980 for definition).
Original entry on oeis.org
1, -1, -1, -1, -2, -4, -10, -25, -66, -178, -490, -1370, -3881, -11113, -32115, -93542, -274332, -809377, -2400641, -7154066, -21409915, -64317898, -193886665, -586311736, -1778101466, -5406660260, -16479943037, -50344990445, -154120149335, -472717222756, -1452529814867, -4470733286364, -13782117172530, -42549485082664, -131545321942331
Offset: 0
-
nmax = 34;
theta0[x_, y_] = Sum[x^n y^(n(n-1)/2), {n, 0, (1/2)(1+Sqrt[1+8nmax]) // Ceiling}];
xi0[y_] = -Sum[a[n] y^n, {n, 0, nmax}];
cc = CoefficientList[theta0[xi0[y], y] + O[y]^(nmax+1) // Normal // Collect[#, y]&, y];
Do[s[n] = Solve[cc[[n+1]] == 0][[1, 1]]; cc = cc /. s[n] , {n, 0, nmax}];
CoefficientList[(-1/xi0[y] /. Array[s, nmax+1, 0]) + O[y]^(nmax+1), y](* Jean-François Alcover, Sep 05 2018 *)
A205999
Inverse Euler transform of A195980.
Original entry on oeis.org
1, 1, 2, 4, 10, 23, 61, 157, 426, 1163, 3253, 9172, 26236, 75634, 220021, 644305, 1898977, 5626720, 16754652, 50104781, 150427938, 453214878, 1369857943, 4152559458, 12621816592, 38459047705, 117453028937, 359455509767, 1102239999454, 3386090204843, 10419804578693, 32115276396739, 99131502581481, 306422345148052, 948423189115351
Offset: 0
-
nmax = 35;
theta0[x_, y_] = Sum[x^n y^(n (n-1)/2), {n, 0, (1/2) (1 + Sqrt[1 + 8 nmax]) // Ceiling}];
xi0[y_] = -Sum[b[n] y^n, {n, 0, nmax}];
cc = CoefficientList[theta0[xi0[y], y] + O[y]^(nmax + 1) // Normal // Collect[#, y]&, y];
Do[s[n] = Solve[cc[[n+1]] == 0][[1, 1]]; cc = cc /. s[n], {n, 0, nmax}];
A195980 = Table[b[n] /. s[n], {n, 1, nmax}];
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i b[[i]] - Sum[c[[d]] b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i) Sum[mob[i, d] c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[A195980] (* Jean-François Alcover, Oct 04 2018 *)
A195982
Coefficients of expansion of 1/xi_0(y)^2 (see A195980 for definition).
Original entry on oeis.org
1, -2, -1, 0, -1, -2, -7, -18, -50, -138, -386, -1092, -3122, -9004, -26173, -76606, -225584, -667880, -1986932, -5936754, -17807936, -53606646, -161892564, -490363820, -1489319219, -4534631182, -13838799043, -42323692348, -129697503097, -398183735878, -1224576726538, -3772166985448, -11637362223230, -35953168834338, -111225021683891
Offset: 0
A206000
Inverse Euler transform of 2 - A195981.
Original entry on oeis.org
1, 0, 0, 1, 2, 6, 15, 40, 110, 303, 853, 2419, 6950, 20110, 58677, 172267, 508825, 1510472, 4504978, 13491155, 40554394, 122318519, 370075767, 1122830663, 3415591776, 10414918850, 31827804005, 97464863424, 299032201886, 919094334980, 2829593105751, 8724968220133, 26942517476931, 83312297368383
Offset: 0
A357233
a(n) = coefficient of x^n in power series A(x) such that: 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
Original entry on oeis.org
1, 1, 3, 11, 46, 207, 980, 4810, 24258, 124951, 654587, 3476985, 18682885, 101372340, 554655435, 3056823864, 16953795008, 94555853982, 529986289496, 2983788539017, 16865736120654, 95677703975144, 544554485912572, 3108656601838926, 17794927199793895
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 207*x^5 + 980*x^6 + 4810*x^7 + 24258*x^8 + 124951*x^9 + 654587*x^10 + 3476985*x^11 + 18682885*x^12 + ...
such that
0 = 1 - A(x) + x*A(x)^3 - x^3*A(x)^6 + x^6*A(x)^10 - x^10*A(x)^15 + x^15*A(x)^21 - x^21*A(x)^28 + ... + (-1)^n*x^(n*(n-1)/2)*A(x)^(n*(n+1)/2) + ...
SPECIFIC VALUES.
A(1/7) = 1.2997111125331190764482142994969231...
A(1/8) = 1.221202992288263902503896694281250380662689...
CONTINUED FRACTION.
The continued fraction in formula (2) may be seen to converge to zero as a limit of successive steps that begin as follows:
[2] 1/(1 + A/(1 - A*(1 - x*A)))
[3] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3)))
[4] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)))))
[5] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5)))))
[6] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5/(1 - x^2*A^3*(1 - x^3*A^3)))))))
...
substituting A = A(x), the resulting power series in x are:
[2] x^2 - 3*x^3 - 13*x^4 - 58*x^5 - 275*x^6 - 1350*x^7 + ...
[3] x^3 - 5*x^4 - 23*x^5 - 111*x^6 - 553*x^7 - 2820*x^8 + ...
[4] x^7 + 11*x^8 + 87*x^9 + 602*x^10 + 3894*x^11 + 24245*x^12 + ...
[5] x^9 + 14*x^10 + 132*x^11 + 1046*x^12 + 7538*x^13 + ...
[6] -x^15 - 21*x^16 - 273*x^17 - 2821*x^18 - 25432*x^19 + ...
[7] -x^18 - 25*x^19 - 375*x^20 - 4375*x^21 - 43800*x^22 + ...
[8] x^26 + 34*x^27 + 663*x^28 + 9725*x^29 + 119226*x^30 + ...
...
the limit of these series converges to zero for |x| < r < 1 where r is the radius of convergence of g.f. A(x).
-
{a(n) = my(A=[1],M=1); for(i=1,n, A = concat(A,0); M = ceil(sqrt(2*(#A)+1));
A[#A] = polcoeff( sum(n=0,M, (-1)^n * x^(n*(n-1)/2) * Ser(A)^(n*(n+1)/2) ), #A-1) ); A[n+1]}
for(n=0,30, print1(a(n),", "))
Showing 1-5 of 5 results.
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