A300673
Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).
Original entry on oeis.org
1, 1, 0, -3, -6, 5, 61, 126, -308, -2772, -5669, 25630, 224730, 486551, -3068155, -29264219, -72173176, 513535711, 5625869262, 16687752839, -113740116822, -1496118902963, -5508392724427, 31534346503605, 523333047780288, 2414704077547660, -10254467367668159
Offset: 0
E.g.f.: A(x) = 1 + x/1! - 3*x^3/3! - 6*x^4/4! + 5*x^5/5! + 61*x^6/6! + 126*x^7/7! - 308*x^8/8! - 2772*x^9/9! - 5669*x^10/10! + ...
Cf.
A008683,
A050385,
A050386,
A068341,
A073776,
A104688,
A117209,
A117210,
A195589,
A204290,
A300663.
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nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[MoebiusMu[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
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a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022
A195588
a(n) = Sum_{k=0..2*n} (-1)^k * moebius(2*n-k+1) * moebius(k+1).
Original entry on oeis.org
1, -3, -1, 2, 3, 1, 1, 6, 0, -3, 5, 2, 1, -1, 5, -8, 5, 10, -7, 2, 3, -9, -1, 6, 4, -3, 8, 2, -3, 3, -11, 2, 13, -15, 5, -2, -5, 5, 13, -8, -7, 9, 3, -2, 18, -1, -7, -4, -14, -6, 7, -4, -3, 2, 1, 6, 7, 7, -9, 18, -13, 7, 14, -12, 1, -7, 1, 0, -3, -13, 7, 6, -3, -5, 22, -16, 3, -1, -11, 2, 8, -5, -15, 6, 1, -9, 3, 18, 1, 10, -13, 8, 9, 3, -15, -2, -7, 6, 16, -4, 1, 1, 3, -2, 49, -7, -9, -6, -1, -9, -3, -20, -13, -11, -11, -22, 12, 25, 7, 0, -6, 5, 3, -2, -18, 4, 7, 4, -1, -7, -5, -2, -15, 3, 32, 2, 15, 11, -1, 12, 5, -23, 3, -2, -17, 1, 10, 4, 7, 16, 13, 34, -2, -31, -11, -12
Offset: 0
G.f.: A(x) = 1 - 3*x - x^2 + 2*x^3 + 3*x^4 + x^5 + x^6 + 6*x^7 +...
where A(x^2) = M(x)*M(-x) and M(x) begins:
M(x) = 1 - x - x^2 - x^4 + x^5 - x^6 + x^9 - x^10 - x^12 + x^13 + x^14 - x^16 +...+ moebius(n+1)*x^n +...
log(A(x)) = -3*x - 11*x^2 - 30*x^3 - 83*x^4 - 243*x^5 - 710*x^6 - 2061*x^7 - 6099*x^8 +...+ -A195589(2*n)*x^n/n +...
Positions of zeros begin:
[8,67,119,161,167,206,207,243,260,263,271,331,339,350,371,407,543,803,...].
Positions of other values of a(n) begin:
+1: [0,5,6,12,54,64,66,84,88,100,101,145,202,210,256,290,309,318,321,...];
-1: [2,13,22,45,77,108,128,138,165,180,216,229,236,348,389,390,418,...];
+2: [3,11,19,27,31,53,79,135,242,360,362,413,548,800,839,...];
-2: [35,43,95,103,123,131,143,152,159,197,235,251,299,324,337,349,...];
+3: [4,20,29,42,76,86,93,102,122,133,142,201,240,326,333,401,518,585,...];
-3: [1,9,25,28,52,68,72,110,166,196,204,234,253,280,340,432,472,653,...];
+4: [24,125,127,147,170,211,269,278,332,459,807,...];
-4: [47,51,99,168,422,538,599,...];
+5: [10,14,16,34,37,121,140,177,308,382,484,520,537,642,645,706,741,...];
-5: [36,73,81,130,173,186,193,217,232,257,302,312,357,373,444,448,...].
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{a(n)=sum(k=0,2*n,(-1)^k*moebius(2*n-k+1)*moebius(k+1))}
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{A195589(n)=n*polcoeff(-log(sum(m=0,n,moebius(m+1)*x^m)+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,-A195589(2*m)*x^m/m)+x*O(x^n)),n)}
A378261
G.f. satisfies A(x) = A(x^2)*M(x)/x, where M(x) = Sum_{n>=1} mu(n)*x^n and mu(n) = A008683(n), the Moebius function of n.
Original entry on oeis.org
1, -1, -2, 1, -2, 3, 3, -2, 0, 1, 5, -3, 1, -6, -2, 5, -5, -1, -1, 1, 10, -3, -18, 4, 0, 6, -2, 2, -7, 3, 10, -8, 3, 13, 3, -4, -5, -1, 16, -7, 1, -21, -21, 13, 11, 32, -3, -18, 14, -12, 27, -5, -29, -14, 40, 18, -9, -7, -28, -9, 28, -12, -1, 21, -18, -17, 21, -4, 2, -26, -55, 15, 6, 57, 76, 4, -63, -35, -31, 12, 27
Offset: 0
G.f.: A(x) = 1 - x - 2*x^2 + x^3 - 2*x^4 + 3*x^5 + 3*x^6 - 2*x^7 + x^9 + 5*x^10 - 3*x^11 + x^12 - 6*x^13 - 2*x^14 + 5*x^15 - 5*x^16 - x^17 - x^18 + x^19 + 10*x^20 - 3*x^21 - 18*x^22 + 4*x^23 + 6*x^25 + ...
where A(x) = A(x^2)*M(x)/x, with M(x) starting as
M(x) = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 - x^17 - x^19 + x^21 + x^22 - x^23 + x^26 - x^29 - x^30 + ... + mu(n)*x^n + ...
so that x = M(x) + M(x^2) + M(x^3) + M(x^4) + ... + M(x^n) + ...
Thus, because M(x) = x*A(x)/A(x^2), we have
x = x*A(x)/A(x^2) + x^2*A(x^2)/A(x^4) + x^3*A(x^3)/A(x^6) + x^4*A(x^4)/A(x^8) + x^5*A(x^5)/A(x^10) + ... + x^n*A(x^n)/A(x^(2*n)) + ...
SPECIFIC VALUES.
A(z) = 0 at z = 0.58029462380732672306477623722678043664946494150129261513733196007593...
at which M(z) = 0; also,
A(+/-sqrt(z)) = 0 where sqrt(z) = 0.76177071603424525645824609227751503049205504631631398431010255922079...
A(t) = 1 at t = -0.36788404155611343004262030413356489215751069093739780571442285347...
and at t = 0.
A(t) = 3/4 at t = 0.1857761904825755826296582919924203148981077141603702423677032...
and at t = -0.490458121286701964509344440548675674917868267307919314675917...
A(t) = 2/3 at t = 0.2335393155019759242193743786104849997659161888659795344146053...
and at t = -0.519733956808611007765606585829046829235954761454838631891973...
A(t) = 1/2 at t = 0.3198405467887820951152560564404574186606511299096542831871690...
and at t = -0.571809669500110081208798541641929396382408973747748977426161...
A(t) = 1/3 at t = 0.3999205691655568373317765145229363880031158016542362401450436...
and at t = -0.620228052536454037811510479582708461188023489655490588138156...
A(t) = 1/4 at t = 0.4397125538469644065749981128335829295493172812091133616367549...
and at t = -0.644814296439482673338297235928747677433880338824104561837150...
A(2/3) = -0.061742613282000323935088569460595490025354693881260621358878068...
A(1/2) = 0.1298319421606939473389436822721728169327823265614184406479845097...
A(1/3) = 0.4724530047506755119306074032801526676340305286513455974985300628...
A(1/4) = 0.6363603730534968288384199665118751095126750453913421545923659477...
A(1/9) = 0.8653204774591832015787933929093873031717584973772619782252127819...
A(1/16) = 0.929904155452109003231349129681442625998610476992091204616469757...
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{a(n) = my(A=1-x, M = sum(m=1,n+1,moebius(m)*x^m) +x^2*O(x^n));
for(i=1,#binary(n)+1, A = subst(A,x,x^2)*M/x ); polcoef(A,n)}
for(n=0,80, print1(a(n),", "))
Showing 1-3 of 3 results.
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