A081543 -id:A081543 - cp's OEIS frontend

A157019 a(n) = Sum_{d|n} binomial(n/d+d-2, d-1).

1, 2, 2, 4, 2, 8, 2, 10, 8, 12, 2, 34, 2, 16, 32, 38, 2, 62, 2, 92, 58, 24, 2, 210, 72, 28, 92, 198, 2, 394, 2, 274, 134, 36, 422, 776, 2, 40, 184, 1142, 2, 1178, 2, 618, 1232, 48, 2, 2634, 926, 1482, 308, 964, 2, 2972, 2004, 4610, 382, 60, 2, 8576, 2, 64, 6470, 5130

Offset: 1

R. J. Mathar, Feb 21 2009

Equals row sums of triangle A156348. - Gary W. Adamson & Mats Granvik, Feb 21 2009
a(n) = 2 iff n is prime.
The binomial transform (note the offset) is 0, 1, 4, 11, 28, 67, 156, 359, 818, 1847, 4146, 9275, ... - R. J. Mathar, Mar 03 2013
a(n) is the number of distinct paths that connect the starting (1,1) point to the hyperbola with equation (x * y = n), when the choice for a move is constrained to belong to { (x := x + 1), (y := y + 1) }. - Luc Rousseau, Jun 27 2017

			a(4) = 4 = 1 + 2 + 0 + 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paul D. Hanna)

Cf. A081543, A018818, A156838 (Mobius transform).
Cf. A156348.
Cf. A000010.

A157019 := proc(n) add( binomial(n/d+d-2, d-1), d=numtheory[divisors](n) ) ; end:

a[n_] := Sum[Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

{a(n)=polcoeff(sum(m=1,n,x^m/(1-x^m+x*O(x^n))^m),n)} \\ Paul D. Hanna, Mar 01 2009

G.f.: A(x) = Sum_{n>=1} x^n/(1 - x^n)^n. - Paul D. Hanna, Mar 01 2009

a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 2, gcd(n,k) - 1) / phi(n/gcd(n,k)) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 2, n/gcd(n,k) - 1) / phi(n/gcd(n,k)) where phi = A000010. - Richard L. Ollerton, May 19 2021

A157020 a(n) = Sum_{d|n} d*binomial(n/d+d-2,d-1).

Original entry on oeis.org

1, 3, 4, 9, 6, 22, 8, 33, 28, 46, 12, 131, 14, 78, 136, 177, 18, 307, 20, 456, 302, 166, 24, 1149, 376, 222, 568, 1177, 30, 2387, 32, 1761, 958, 358, 2556, 5224, 38, 438, 1496, 7851, 42, 8317, 44, 4863, 9136, 622, 48, 20169, 6518, 11451, 3112, 8516, 54, 23734

Offset: 1

R. J. Mathar, Feb 21 2009

Equals row sums of triangle A157497. [Gary W. Adamson & Mats Granvik, Mar 01 2009]

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

Cf. A081543, A132065, A156833 (Mobius transform), A324158, A324159.

add( d*binomial(n/d+d-2,d-1),d=numtheory[divisors](n) ) ;

N=66; x='x+O('x^N); Vec(sum(k=1, N, k*x^k/(1-x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

G.f.: Sum_{n>=1} n*x^n/(1-x^n)^n.

A338662 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 5, 10, 29, 26, 123, 50, 305, 352, 668, 122, 3844, 170, 2593, 9704, 13825, 290, 41598, 362, 118259, 107986, 33047, 530, 929102, 394376, 130744, 1203580, 2737415, 842, 9910225, 962, 13315073, 14199222, 2404670, 33547310, 136502007, 1370, 10555795, 168405072, 548460064, 1682

Offset: 1

Seiichi Manyama, Apr 22 2021

nonn

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

Cf. A081543, A338663, A343574.

a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Apr 22 2021 *)

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

N=66; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-k*x^k)^k-1))

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

If p is prime, a(p) = 1 + p^2.

A338682 a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 1, 4, 0, 6, 3, 8, -8, 20, 0, 12, -12, 14, -7, 72, -65, 18, 10, 20, -61, 142, -33, 24, -203, 152, -52, 248, -183, 30, 121, 32, -617, 398, -102, 828, -619, 38, -133, 600, -896, 42, 140, 44, -870, 2864, -207, 48, -4438, 1766, 751, 1192, -1587, 54, -348, 4424, -3011, 1598, -348, 60

Offset: 1

Seiichi Manyama, Apr 23 2021

sign

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

Cf. A081543, A217670, A318636, A338683, A338684.

a[n_] := DivisorSum[n, (-1)^(# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* Amiram Eldar, Apr 24 2021 *)

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

N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+x^k)^k))

N=66; x='x+O('x^N); Vec(-sum(k=1, N, (-x)^k/(1-x^k)^(k+1)))

G.f.: Sum_{k >= 1} (1 - 1/(1 + x^k)^k).
G.f.: - Sum_{k >= 1} (-x)^k/(1 - x^k)^(k+1).
If p is prime, a(p) = (-1)^(p-1) + p.

A338663 a(n) = Sum_{d|n} (n/d)^n * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 9, 82, 1073, 15626, 284567, 5764802, 134874369, 3486981232, 100146490520, 3138428376722, 107039261352736, 3937376385699290, 155587085803983069, 6568409424129452048, 295158038428838854657, 14063084452067724991010, 708242105301294465144506, 37589973457545958193355602

Offset: 1

Seiichi Manyama, Apr 22 2021

nonn

Cf. A081543, A338662.

a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Apr 22 2021 *)

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

N=20; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-(k*x)^k)^k-1))

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

If p is prime, a(p) = 1 + p^(p+1).

A343574 a(n) = Sum_{d|n} d^d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 6, 30, 272, 3130, 46794, 823550, 16778544, 387420768, 10000018820, 285311670622, 8916100779324, 302875106592266, 11112006832146486, 437893890380925960, 18446744073860555680, 827240261886336764194, 39346408075300413088392

Offset: 1

Seiichi Manyama, Apr 20 2021

nonn

Cf. A081543, A343568, A343573.

a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-x^k)^(k+1)))

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

If p is prime, a(p) = p + p^p.

A338658 a(n) = Sum_{d|n} d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 4, 6, 14, 10, 36, 14, 56, 48, 80, 22, 228, 26, 140, 240, 316, 34, 552, 38, 820, 546, 308, 46, 2088, 680, 416, 1044, 2156, 58, 4380, 62, 3248, 1782, 680, 4690, 9672, 74, 836, 2808, 14560, 82, 15456, 86, 9108, 17040, 1196, 94, 37704, 12110, 21420, 5916, 16068, 106, 44496

Offset: 1

Seiichi Manyama, Apr 22 2021

nonn

Cf. A081543, A343574.

a[n_] := DivisorSum[n, # * Binomial[# + n/# - 1, #] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-x^k)^(k+1)))

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

If p is prime, a(p) = 2*p.

A360794 Expansion of Sum_{k>0} x^k / (1 - k * x^k)^(k+1).

Original entry on oeis.org

1, 3, 4, 11, 6, 43, 8, 109, 100, 281, 12, 1507, 14, 1863, 3376, 6937, 18, 26245, 20, 53211, 63022, 67739, 24, 572413, 78776, 372945, 1087048, 1761719, 30, 7362871, 32, 9947953, 16897486, 10027349, 8011116, 123101515, 38, 49807779, 241823440, 361722421, 42

Offset: 1

Seiichi Manyama, Feb 21 2023

nonn

Cf. A360782, A360783.

Cf. A081543, A324158.

a[n_] := DivisorSum[n, #^(n/# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Jul 31 2023 *)

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-k*x^k)^(k+1)))

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

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

If p is prime, a(p) = 1 + p.

A340626 a(n) = Sum_{d|n, d odd} binomial(d+n/d-1, d).

Original entry on oeis.org

1, 2, 4, 4, 6, 10, 8, 8, 20, 16, 12, 32, 14, 22, 72, 16, 18, 84, 20, 76, 142, 34, 24, 144, 152, 40, 248, 148, 30, 518, 32, 32, 398, 52, 828, 620, 38, 58, 600, 832, 42, 1416, 44, 408, 2864, 70, 48, 864, 1766, 2078, 1192, 612, 54, 3224, 4424, 3488, 1598, 88, 60, 6784, 62, 94, 13528, 64, 8634

Offset: 1

Seiichi Manyama, Apr 25 2021

nonn

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

Cf. A081543, A338682, A340625.

a[n_] := DivisorSum[n, Binomial[# + n/# - 1, #] &, OddQ[#] &]; Array[a, 65] (* Amiram Eldar, Apr 25 2021 *)

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

N=66; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-x^k)^k-1/(1+x^k)^k)/2)

N=66; x='x+O('x^N); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(2*k-1))^(2*k)))

G.f.: (1/2) * Sum_{k >= 1} (1/(1 - x^k)^k - 1/(1 + x^k)^k).
G.f.: Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1))^(2*k).
a(n) = (A081543(n) + A338682(n))/2.
If p is prime, a(p) = (p mod 2) + p.

A344777 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, -1, 4, -6, 6, -3, 8, -22, 20, 0, 12, -44, 14, 7, 72, -95, 18, -10, 20, -71, 142, 33, 24, -399, 152, 52, 248, -57, 30, -121, 32, -679, 398, 102, 828, -685, 38, 133, 600, -1568, 42, -140, 44, 318, 2864, 207, 48, -5858, 1766, -751, 1192, 831, 54, 348, 4424, -3979, 1598, 348, 60

Offset: 1

Seiichi Manyama, May 28 2021

sign

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

Cf. A081543, A217670, A338682.

a[n_] := DivisorSum[n, (-1)^(n/# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* Amiram Eldar, May 28 2021 *)

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+x^k)^(k+1)))

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

If p is prime, a(p) = 1 + (-1)^(p-1) * p.

cp's OEIS Frontend

A157019 a(n) = Sum_{d|n} binomial(n/d+d-2, d-1).

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A157020 a(n) = Sum_{d|n} d*binomial(n/d+d-2,d-1).

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A338662 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n/d-1, d).

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A338682 a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).

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A338663 a(n) = Sum_{d|n} (n/d)^n * binomial(d+n/d-1, d).

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A343574 a(n) = Sum_{d|n} d^d * binomial(d+n/d-1, d).

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PARI

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A338658 a(n) = Sum_{d|n} d * binomial(d+n/d-1, d).

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A360794 Expansion of Sum_{k>0} x^k / (1 - k * x^k)^(k+1).

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