A048272 -id:A048272 - cp's OEIS frontend

A363630 Expansion of Sum_{k>0} (1/(1+x^k)^3 - 1).

-3, 3, -13, 18, -24, 21, -39, 63, -68, 48, -81, 127, -108, 87, -170, 216, -174, 156, -213, 294, -302, 201, -303, 497, -375, 276, -474, 537, -468, 426, -531, 777, -686, 462, -726, 965, -744, 573, -938, 1200, -906, 798, -993, 1251, -1306, 831, -1179, 1875, -1314, 1023, -1562, 1722, -1488, 1290, -1698

Offset: 1

Seiichi Manyama, Jun 12 2023

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

Cf. A002129, A048272, A321543, A363629, A363631.

Cf. A320900, A363022, A363615.

a[n_] := DivisorSum[n, (-1)^#*Binomial[# + 2, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)

a(n) = sumdiv(n, d, (-1)^d*binomial(d+2, 2));

G.f.: Sum_{k>0} binomial(k+2,2) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+2,2).
a(n) = -(A321543(n) + 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025

A295794 Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).

Original entry on oeis.org

1, 1, 1, 13, 25, 241, 2761, 14701, 153553, 1903105, 27877681, 263555821, 4788201001, 65083782193, 1040877257785, 24098794612621, 373918687272481, 7393663746307201, 164894196647876833, 3504497611085823565, 81863829346282866361, 2257321249626793901041, 49755091945025205954601

Offset: 0

Ilya Gutkovskiy, Nov 27 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A028342, A048272, A168243, A206303, A294363, A294392, A295739.

a:=series(mul(exp(x^k/(1+x^k)),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

nmax = 22; CoefficientList[Series[Product[Exp[x^k/(1 + x^k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x D[Log[Product[(1 + x^k)^(1/k), {k, 1, nmax}]], x]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[-k Sum[(-1)^d, {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} A048272(k)*x^k).
E.g.f.: exp(x*f'(x)), where f(x) = log(Product_{k>=1} (1 + x^k)^(1/k)).
a(n) ~ exp(2*sqrt(n*log(2)) - 1/4 - n) * n^(n - 1/4) * log(2)^(1/4) / sqrt(2). - Vaclav Kotesovec, Sep 07 2018

A300187 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k)^(n/k).

Original entry on oeis.org

1, 1, 4, 39, 488, 7615, 147024, 3371137, 89079808, 2665537713, 89142430400, 3295096700071, 133399600068096, 5870116973678191, 278971698167158528, 14239859507270510625, 776985219329347518464, 45130494178637796970273, 2780224621391401396134912, 181059775626543107582734183

Offset: 0

Ilya Gutkovskiy, Feb 28 2018

nonn

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 + x^k)^(n/k) begins:
n = 0: (1), 0,   0,    0,     0,      0,       0,  ...
n = 1:  1, (1),  1,    5,    11,     59,     439,  ...
n = 2:  1,  2,  (4),  16,    68,    328,    2416,  ...
n = 3:  1,  3,   9,  (39),  207,   1197,    8811,  ...
n = 4:  1,  4,  16,   80,  (488),  3296,   25984,  ...
n = 5:  1,  5,  25,  145,   995,  (7615),  65575,  ...
n = 6:  1,  6,  36,  240,  1836,  15624, (147024), ...

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

Cf. A048272, A168243, A270922, A299033, A299034, A300188.

Table[n! SeriesCoefficient[Product[(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

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

a(n) ~ c * d^n * n^n, where d = 1.294982800733109500251... and c = 0.6755467963500480915... - Vaclav Kotesovec, Sep 07 2018

A321544 a(n) = Sum_{d|n} (-1)^(d-1)*d^5.

Original entry on oeis.org

1, -31, 244, -1055, 3126, -7564, 16808, -33823, 59293, -96906, 161052, -257420, 371294, -521048, 762744, -1082399, 1419858, -1838083, 2476100, -3297930, 4101152, -4992612, 6436344, -8252812, 9768751, -11510114, 14408200, -17732440, 20511150, -23645064, 28629152, -34636831, 39296688, -44015598

Offset: 1

N. J. A. Sloane, Nov 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Divisor sums Sum_{d|n} (-1)^(d-1)*d^k: A048272 (k = 0), A002129 (k = 1), A321543 (k = 2), A138503 (k = 3), A279395 (k = 4, unsigned), A321545 - A321551 (k = 6 to k = 12).

Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^5, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); f[2, e_] := 2 - (2^(5*e + 5) - 1)/31; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^5), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^5*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(x^(4*n) - 26*x^(3*n) + 66*x^(2*n) - 26*x^n + 1)/(1 + x^n)^6 (note [1,26,66,26,1] is row 5 of A008292). - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(5*e + 5) - 1)/31, and a(p^e) = (p^(5*e + 5) - 1)/(p^5 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 2, 5, 3, 4, 3, 6, 5, 6, 3, 7, 6, 7, 6, 9, 6, 7, 6, 11, 9, 10, 7, 10, 9, 10, 9, 14, 11, 12, 9, 13, 12, 13, 10, 15, 14, 15, 14, 17, 12, 13, 12, 19, 17, 18, 15, 18, 17, 18, 15, 20, 17, 18, 17, 22, 21, 22, 17, 23, 20, 21, 20, 23, 20, 21, 20, 27, 26, 27, 22, 25, 22, 23, 22

Offset: 1

Ilya Gutkovskiy, May 25 2020

nonn

a(n) = number of terms among {ceiling(n/k)}, 1 <= k <= n, that are odd.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002541, A006218, A048272, A059851, A075997, A120885, A320226, A325939, A332682, A333194.

b:= n-> add((-1)^d, d=numtheory[divisors](n)):
a:= proc(n) option remember; `if`(n>0, 1+b(n-1)+a(n-1), 0) end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 25 2020

Table[Sum[Mod[Ceiling[n/k], 2], {k, 1, n}], {n, 1, 80}]
Table[n - Sum[DivisorSum[k, (-1)^(# + 1) &], {k, 1, n - 1}], {n, 1, 80}]
nmax = 80; CoefficientList[Series[x/(1 - x) (1 + Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, ceil(n/k) % 2); \\ Michel Marcus, May 25 2020

from math import isqrt
def A330926(n): return n+(s:=isqrt(n-1))**2-((t:=isqrt(m:=n-1>>1))**2<<1)-(sum((n-1)//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} x^(2*k) / (1 + x^k)).
a(n) = n - Sum_{k=1..n-1} A048272(k).
a(n) = A075997(n-1) + 1.

A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A038548, A048272, A113652, A228441, A305152, A333781, A348660, A348951, A348952, A258998, A333809.

Table[DivisorSum[n, (-1)^(n/# + 1) &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A348515(n) = sumdiv(n,d,if((d*d)<=n,(-1)^(1 + (n/d)),0)); \\ Antti Karttunen, Nov 05 2021

from sympy import divisors
def a(n): return sum((-1)**(n//d + 1) for d in divisors(n) if d*d <= n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 22 2021

G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k^2) / (1 + x^k).
a(n) = 1 iff n = 1 or n is an odd prime (A006005). - Bernard Schott, Nov 22 2021
a(n) = A258998(n) - A348951(n). - Ridouane Oudra, Aug 21 2025
a(n) = A048272(n) - A333809(n). - Ridouane Oudra, Sep 01 2025

A354506 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) )/(k * (n-k)!).

Original entry on oeis.org

1, 2, 7, 14, 63, 284, 2385, 3940, 87717, 940126, 12743267, 30055618, 562302323, 9005878920, 423435780989, 2080544097000, 24457758561001, 444510436079706, 17533073308723423, 46973556239255702, 7501223613055891783, 178483805340054632084, 4396051786608296882889, -31788150263554644516724

Offset: 1

Seiichi Manyama, Aug 15 2022

sign

Cf. A354507, A354508.

Cf. A048272, A356389.

a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/(k*(n-k)!));

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, log(1+x^k)/k)))

a(n) = n! * Sum_{k=1..n} A048272(k)/(k * (n-k)!).

E.g.f.: exp(x) * Sum_{k>0} log(1 + x^k)/k.

A356564 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^x.

Original entry on oeis.org

1, 0, 2, 0, 28, -30, 888, -1260, 51728, -196560, 5293080, -22286880, 710229408, -4851269280, 138348035616, -1091188098000, 36482139114240, -379928382462720, 11812558481332992, -137793570801143040, 4609972759421554560, -67292912045817561600

Offset: 0

Seiichi Manyama, Aug 12 2022

sign

Cf. A356565, A356566.

Cf. A007113, A048272, A338814, A356392.

my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k))^x))

a048272(n) = sumdiv(n, d, (-1)^(n/d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a048272(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * A048272(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A363629 Expansion of Sum_{k>0} (1/(1+x^k)^2 - 1).

Original entry on oeis.org

-2, 1, -6, 6, -8, 4, -10, 15, -16, 6, -14, 22, -16, 8, -28, 32, -20, 13, -22, 32, -36, 12, -26, 56, -34, 14, -44, 42, -32, 24, -34, 65, -52, 18, -52, 68, -40, 20, -60, 82, -44, 32, -46, 62, -84, 24, -50, 122, -60, 31, -76, 72, -56, 40, -76, 108, -84, 30, -62, 124, -64, 32, -110, 130, -88, 48, -70, 92

Offset: 1

Seiichi Manyama, Jun 12 2023

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

Cf. A363630, A363631.

Cf. A002129, A048272, A325940.

a[n_] := DivisorSum[n, (-1)^#*(# + 1) &]; Array[a, 100] (* Amiram Eldar, Jul 18 2023 *)

PARI
```
a(n) = sumdiv(n, d, (-1)^d*(d+1));
```

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

a(n) = Sum_{d|n} (-1)^d * (d+1) = -(A002129(n) + A048272(n)).

A364204 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 14 2023

sign

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

Cf. A001817, A048272, A364011, A364205, A364232.

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

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

cp's OEIS Frontend

A363630 Expansion of Sum_{k>0} (1/(1+x^k)^3 - 1).

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A295794 Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).

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A300187 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k)^(n/k).

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A321544 a(n) = Sum_{d|n} (-1)^(d-1)*d^5.

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A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

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A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

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A354506 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) )/(k * (n-k)!).

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A356564 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^x.

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A364204 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1)).