A048272 -id:A048272 - cp's OEIS frontend

A364205 Expansion of Sum_{k>=0} x^(3k+2) / (1 + x^(3k+2)).

0, 1, 0, -1, 1, 1, 0, 0, 0, 0, 1, -1, 0, 2, 1, -2, 1, 1, 0, -1, 0, 0, 1, 0, 1, 2, 0, -2, 1, 0, 0, -1, 1, 0, 2, -1, 0, 2, 0, -2, 1, 2, 0, -1, 1, 0, 1, -2, 0, 1, 1, -2, 1, 1, 2, 0, 0, 0, 1, -1, 0, 2, 0, -3, 2, 0, 0, -1, 1, 0, 1, 0, 0, 2, 1, -2, 2, 2, 0, -3, 0, 0, 1, -2, 2, 2, 1, -2, 1, 0, 0, -1

Offset: 1

Ilya Gutkovskiy, Jul 14 2023

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Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001822, A048272, A364014, A364204, A364235.

nmax = 92; CoefficientList[Series[Sum[x^(3 k + 2)/(1 + x^(3 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 92}]

a(n) = Sum_{d|n, n/d==2 (mod 3)} (-1)^(d+1).

A368929 Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).

Original entry on oeis.org

1, -1, 17, -16, 49, -17, 97, -112, 225, -49, 241, -272, 337, -97, 833, -640, 577, -225, 721, -784, 1649, -241, 1057, -1904, 1825, -337, 2673, -1552, 1681, -833, 1921, -3328, 4097, -577, 4753, -3600, 2737, -721, 5729, -5488, 3361, -1649, 3697, -3856, 11025, -1057, 4417

Offset: 1

Vaclav Kotesovec, Jan 12 2024

Dirichlet convolution of A007434 and A162395.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048272, A332794.

Cf. A007434, A162395, A360428.

Table[Sum[Sum[d^2 * MoebiusMu[k/d], {d, Divisors[k]}] * (-1)^(n/k + 1) * n^2/k^2, {k, Divisors[n]}], {n, 1, 100}]
f[p_, e_] := p^(2*e)*(1 + e*(1 - 1/p^2)); f[2, e_] := -(3*e - 2)*2^(2*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2024 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, -(3*e-2)*2^(2*e-2), p^(2*e)*(1 + e*(1-1/p^2))));} \\ Amiram Eldar, Jan 12 2024

Sum_{k=1..n} a(k) ~ log(2) * n^3 / (3*zeta(3)).

Multiplicative with a(2^e) = -(3*e-2)*2^(2*e-2), and a(p^e) = p^(2*e)*(1 + e*(1-1/p^2)) for an odd prime p. - Amiram Eldar, Jan 12 2024

A099751 Number of ways to write n as differences of (-4)-gonal numbers. If pe(n):=1/2n((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 2, 3, 0, 1, 2, 2, 0, 2, 4, 2, 0, 4, 1, 2, 0, 2, 4, 2, 0, 2, 2, 2, 0, 2, 3, 3, 0, 2, 2, 2, 0, 4, 4, 2, 0, 2, 2, 2, 0, 2, 5, 4, 0, 2, 2, 2, 0, 2, 2, 2, 0, 3, 2, 4, 0, 2, 6, 1, 0, 2, 2, 4, 0, 2, 4, 2, 0, 4, 2, 2, 0, 4, 4, 2, 0, 2, 3, 2, 0, 2, 4, 4

Offset: 1

Volker Schmitt (clamsi(AT)gmx.net), Nov 10 2004

			G.f. = x + x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^8 + x^9 + 2*x^11 + x^12 + 2*x^13 + ...
a(5)=2 because there are two ways of differences: First pe(3)-pe(-2)=(-15)-(-20)=5 and second pe(1)-pe(2)=(1)-(-4)=5, for e=-4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001227 for e in {3, -2, 6}, A048272 for e in {0, 1, 4, 8} and A035218 for e=-1.

Cf. A001620, A027748, A124010, A256232, A035191.

a099751 n = product $ zipWith f (a027748_row n) (a124010_row n)
   where f 2 e = e - 1; f 3 e = 1; f _ e = e + 1
-- Reinhard Zumkeller, Mar 20 2015

res:=1; ifac:=op(ifactors(i))[2]; for pfac in ifac do; if pfac[1]=2 then res:=res*(pfac[2]-1); else if pfac[1]<>3 then res:=res*(pfac[2]+1); fi; fi; od; a(i):=res;

a[ n_] := If[ n < 1, 0, If[ Divisible[n, 4], -1, 1] Sum[ KroneckerSymbol[ -3, d] (-1)^Quotient[ d, 3], {d, Divisors@n}]]; (* Michael Somos, Mar 19 2015 *)

{a(n) = if( n<1, 0, if( n%2==0, (valuation(n, 2) -1) * a(n / 2^valuation(n, 2)), if( n%3==0, a(n / 3^valuation(n, 3)), numdiv(n)))) }; /* Michael Somos, Sep 20 2005 */

{a(n) = if( n<1, 0, (-1)^(n%4 == 0) * sumdiv( n, d, (-1)^(d\3) * kronecker( -3, d)))}; /* Michael Somos, Nov 16 2011 */

Multiplicative with a(2^e) = e-1 if e>0, a(3^e) = 1, a(p^e) = e+1 if p>3.
Moebius transform is period 12 sequence [ 1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, ...].
G.f.: Sum_{k>0} (x^k - x^(2*k) + x^(4*k) + x^(5*k) + x^(7*k) + x^(8*k) - x^(10*k) + x^(11*k)) / (1 - x^(12*k)). - Michael Somos, Sep 20 2005
a(3*n) = a(n). a(4*n + 2) = 0. - Michael Somos, Nov 16 2011
a(4*n) = A035191(n). - Michael Somos, Mar 19 2015
From Amiram Eldar, Nov 30 2022: (Start)
Dirichlet g.f.: zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s))*(1 - 1/3^s).
Sum_{k=1..n} a(k) ~ n*log(n)/3 + (2*gamma - 1 + log(3)/2)*n/3, where gamma is Euler's constant (A001620). (End)

A258453 G.f.: Sum_{k>0} x^((k^2 + k)/2) / (1 + x^k).

Original entry on oeis.org

1, -1, 2, -1, 0, 0, 2, -1, -1, 0, 2, 0, 0, -2, 2, -1, 0, 1, 2, -2, 0, -2, 2, 0, 1, 0, 0, 0, 0, -2, 2, -1, 0, 0, 2, 1, 0, -2, 0, -2, 0, 2, 2, -2, 2, -2, 2, 0, -1, -1, 0, 0, 0, -2, 4, 0, 0, 0, 2, -2, 0, -2, 0, -1, 0, 2, 2, 0, 0, -2, 2, -1, 0, 0, 2, -2, -2, 0, 2

Offset: 1

Michael Somos, Nov 05 2015

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			G.f. = x - x^2 + 2*x^3 - x^4 + 2*x^7 - x^8 - x^9 + 2*x^11 - 2*x^14 + 2*x^15 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A001227, A048272.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^((k^2 + k)/2) / (1 + x^k), {k, Sqrt[8 n + 1]}], {x, 0, n}]];

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(8*n + 1), x^((k^2 + k)/2) / (1 + x^k), x * O(x^n)), n))};

A300188 a(n) = n! * [x^n] Product_{k>=1} 1/(1 + x^k)^(n/k).

Original entry on oeis.org

1, -1, 4, -39, 536, -9115, 185904, -4461877, 123647488, -3886461081, 136538590400, -5300491027711, 225313697972736, -10409021924850211, 519298241645107456, -27824560148201248125, 1593597443825288904704, -97153909607626767338353, 6281720886474120790582272

Offset: 0

Ilya Gutkovskiy, Feb 28 2018

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			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 + x^k)^(n/k) begins:
n = 0: (1),  0,   0,     0,     0,       0,        0,  ...
n = 1:  1, (-1),  1,    -5,    23,    -119,      619,  ...
n = 2:  1,  -2,  (4),  -16,    92,    -568,     3856,  ...
n = 3:  1,  -3,   9,  (-39),  243,   -1737,    13671,  ...
n = 4:  1,  -4,  16,   -80,  (536),  -4256,    37504,  ...
n = 5:  1,  -5,  25,  -145,  1055,  (-9115),   88075,  ...
n = 6:  1,  -6,  36,  -240,  1908,  -17784,  (185904), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A048272, A281266, A294356, A299033, A299034, A300187.

Table[n! SeriesCoefficient[Product[1/(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]

a(n) = n! * [x^n] exp(-n*Sum_{k>=1} A048272(k)*x^k/k).

a(n) ~ (-1)^n * c * d^n * n^n, where d = 1.3587950730244927060955... and c = 0.6449711831436950784... - Vaclav Kotesovec, Sep 08 2018

A307396 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k*A(x)^k/(1 + x^k).

Original entry on oeis.org

1, 1, 1, 4, 9, 25, 78, 235, 734, 2355, 7637, 25096, 83394, 279563, 944559, 3213254, 10996236, 37829956, 130759164, 453879479, 1581472334, 5529435704, 19393856909, 68217376618, 240586328527, 850553637256, 3013750513593, 10700805837614, 38068482070675, 135674217800041

Offset: 0

Ilya Gutkovskiy, Apr 07 2019

nonn

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 25*x^5 + 78*x^6 + 235*x^7 + 734*x^8 + 2355*x^9 + 7637*x^10 + ...

Cf. A048272, A192207, A192401, A307398, A307400.

terms = 30; A[] = 0; Do[A[x] = 1 + Sum[x^k A[x]^k /(1 + x^k), {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 30; A[] = 0; Do[A[x] = 1 + Sum[x^k Sum[(-1)^(k/d + 1) A[x]^d, {d, Divisors[k]}], {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k * Sum_{d|k} (-1)^(k/d+1)*A(x)^d.

A317528 Expansion of Sum_{k>=1} mu(k)^2*x^k/(1 + x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 2, -2, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 4, -2, 2, 0, 2, -4, 4, 0, 2, -4, 2, 0, 2, -4, 2, 0, 2, -2, 4, 0, 4, -4, 2, 0, 4, -4, 2, 0, 2, -4, 4, 0, 2, -4, 2, 0, 4, -4, 2, 0, 4, -4, 4, 0, 2, -8, 2, 0, 4, -2, 4, 0, 2, -4, 4, 0, 2, -4, 2, 0, 4, -4, 4, 0, 2, -4, 2, 0, 2, -8, 4, 0, 4, -4, 2, 0, 4, -4, 4, 0, 4

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005117, A008683, A008966, A034444, A048272, A100007, A300894, A305614.

with(numtheory): seq(coeff(series(add(mobius(k)^2*x^k/(1+x^k),k=1..n), x,n+1),x,n),n=1..120); # Muniru A Asiru, Jul 30 2018

nmax = 95; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + MoebiusMu[k]^2 x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, SquareFreeQ[#] &], {n, 95}]
f[p_, e_] := 2; f[2, e_] := If[e == 1, 0, -2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2022 *)

A317528(n) = sumdiv(n,d,((-1)^(1+d))*issquarefree(n/d)); \\ Antti Karttunen, Dec 05 2021

G.f.: Sum_{k>=1} x^A005117(k)/(1 + x^A005117(k)).
L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A008966(d).
If n is odd, a(n) = A034444(n).
Multiplicative with a(2) = 0, a(2^e) = -2 for e>1, and a(p^e) = 2 for p>2 and e>=1. - Amiram Eldar, Nov 19 2022

A317531 Expansion of Sum_{p prime, k>=1} x^(p^k)/(1 + x^(p^k)).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 2, -2, 1, -1, 1, -1, 2, 0, 1, -2, 2, 0, 3, -1, 1, -1, 1, -3, 2, 0, 2, -2, 1, 0, 2, -2, 1, -1, 1, -1, 3, 0, 1, -3, 2, -1, 2, -1, 1, -2, 2, -2, 2, 0, 1, -2, 1, 0, 3, -4, 2, -1, 1, -1, 2, -1, 1, -3, 1, 0, 3, -1, 2, -1, 1, -3, 4, 0, 1, -2, 2, 0, 2, -2, 1, -2, 2, -1, 2, 0, 2

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

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Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001222, A048272, A069513, A246655, A288571, A305614, A317528.

nmax = 95; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + Boole[PrimePowerQ[k]] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, PrimePowerQ[#] &], {n, 95}]

A317531(n) = sumdiv(n,d,((-1)^(n/d+1))*(1==omega(d))); \\ Antti Karttunen, Sep 30 2018

G.f.: Sum_{k>=1} x^A246655(k)/(1 + x^A246655(k)).
L.g.f.: log(Product_{p prime, k>=1} (1 + x^(p^k))^(1/p^k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A069513(d).
If n is odd, a(n) = A001222(n).

A327274 Dirichlet g.f.: 1 / (zeta(s)^2 * (1 - 2^(1 - s))).

Original entry on oeis.org

1, 0, -2, 1, -2, 0, -2, 2, 1, 0, -2, -2, -2, 0, 4, 4, -2, 0, -2, -2, 4, 0, -2, -4, 1, 0, 0, -2, -2, 0, -2, 8, 4, 0, 4, 1, -2, 0, 4, -4, -2, 0, -2, -2, -2, 0, -2, -8, 1, 0, 4, -2, -2, 0, 4, -4, 4, 0, -2, 4, -2, 0, -2, 16, 4, 0, -2, -2, 4, 0, -2, 2, -2, 0, -2, -2, 4, 0, -2, -8

Offset: 1

Ilya Gutkovskiy, Oct 22 2019

Dirichlet inverse of A048272.

Moebius transform of A067856.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007427, A008683, A048272, A062503 (positions of 1's), A067856, A327268.

a[1] = 1; a[n_] := Sum[Sum[(-1)^j, {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 80}]
f[p_, e_] := Switch[e, 1, -2, 2, 1, , 0]; f[2, e] := 2^(e-2); f[2, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

A067856(n) = { my(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1)); }; \\ From A067856
A327274(n) = sumdiv(n,d,moebius(n/d)*A067856(d));

a(1) = 1; a(n) = -Sum_{d|n, dA048272(n/d) * a(d).
a(n) = Sum_{d|n} mu(n/d) * A067856(d).
a(n) = 0 if n == 2 (mod 4). - Bernard Schott, Dec 07 2021
Multiplicative with a(2) = 0, a(2^e) = 2^(e-2) for e >= 2, and for an odd prime p, a(p) = -2, a(p^2) = 1, and a(p^e) = 0 for e >= 3. - Amiram Eldar, Sep 15 2023

A329801 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 + x^(k(k + 1)/2)).

Original entry on oeis.org

1, -1, 2, -1, 1, -1, 1, -1, 2, 0, 1, -3, 1, -1, 3, -1, 1, -1, 1, -2, 3, -1, 1, -3, 1, -1, 2, 0, 1, -1, 1, -1, 2, -1, 1, -2, 1, -1, 2, -2, 1, -2, 1, -1, 4, -1, 1, -3, 1, 0, 2, -1, 1, -1, 2, -2, 2, -1, 1, -5, 1, -1, 3, -1, 1, 0, 1, -1, 2, 0, 1, -4, 1, -1, 3, -1, 1, 0, 1, -2, 2, -1, 1, -3, 1

Offset: 1

Ilya Gutkovskiy, Nov 21 2019

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Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000217, A007862, A010054, A048272, A304876, A317529.

nmax = 85; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d + 1) Boole[IntegerQ[Sqrt[8 d + 1]]], {d, Divisors[n]}], {n, 1, 85}]

A329801(n) = sumdiv(n,d,((-1)^(1+(n/d))) * ispolygonal(d,3)); \\ Antti Karttunen, Jan 15 2025

G.f.: Sum_{k>=1} (-1)^(k + 1) * theta_2(x^(k/2)) / (2 * x^(k/8)).

a(n) = Sum_{d|n} (-1)^(n/d + 1) * A010054(d).

cp's OEIS Frontend

A364205 Expansion of Sum_{k>=0} x^(3*k+2) / (1 + x^(3*k+2)).

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A368929 Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).

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A099751 Number of ways to write n as differences of (-4)-gonal numbers. If pe(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4.

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A258453 G.f.: Sum_{k>0} x^((k^2 + k)/2) / (1 + x^k).

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A300188 a(n) = n! * [x^n] Product_{k>=1} 1/(1 + x^k)^(n/k).

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A307396 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k*A(x)^k/(1 + x^k).

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A317528 Expansion of Sum_{k>=1} mu(k)^2*x^k/(1 + x^k), where mu() is the Moebius function (A008683).

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A317531 Expansion of Sum_{p prime, k>=1} x^(p^k)/(1 + x^(p^k)).

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A364205 Expansion of Sum_{k>=0} x^(3k+2) / (1 + x^(3k+2)).

A099751 Number of ways to write n as differences of (-4)-gonal numbers. If pe(n):=1/2n((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4.

A329801 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 + x^(k(k + 1)/2)).