A332470 -id:A332470 - cp's OEIS frontend

A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

1, 3, 7, 25, 71, 280, 925, 3561, 12916, 49346, 184757, 710255, 2704157, 10427747, 40119781, 155288897, 601080391, 2334714319, 9075135301, 35352181116, 137846759282, 538302226628, 2104098963721, 8233718962365, 32247603703576, 126412458920775, 495918551104687

Offset: 1

Ilya Gutkovskiy, Feb 14 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A308814, A332470.

Table[DivisorSum[n, Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]

a(n) = sumdiv(n, d, binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

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

a(n) ~ 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 04 2022

A343553 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).

Original entry on oeis.org

1, 3, 8, 26, 74, 287, 930, 3572, 12966, 49379, 184766, 710712, 2704168, 10427822, 40123208, 155289768, 601080406, 2334740919, 9075135318, 35352194658, 137846990678, 538302226835, 2104098963742, 8233721100024, 32247603765020, 126412458921072, 495918569262798

Offset: 1

Seiichi Manyama, Apr 19 2021

nonn

			a(3) = gcd(1,1,3) + gcd(1,2,3) + gcd(1,3,3) + gcd(2,2,3) + gcd(2,3,3) + gcd(3,3,3) = 1 + 1 + 1 + 1 + 1 + 3 = 8.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000010, A332470, A332508, A343516, A343517, A343547.

a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + n - 2, n-1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)

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

a(n) = A343516(n,n-1).
a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-2, n-1).
a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^n.
a(n) ~ 2^(2*n - 2) / sqrt(Pi*n). - Vaclav Kotesovec, May 23 2021

A177975 Square array T(n,k) read by antidiagonals up. Each column is the first column in the matrix inverse of a triangular matrix that is the k-th differences of A051731 in the column direction.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 7, 9, 4, 1, 0, 2, 14, 16, 14, 5, 1, 0, 6, 13, 34, 30, 20, 6, 1, 0, 4, 27, 43, 69, 50, 27, 7, 1, 0, 6, 26, 83, 107, 125, 77, 35, 8, 1, 0, 4, 39, 100, 209, 226, 209, 112, 44, 9, 1, 0, 10, 38, 155, 295, 461, 428, 329, 156, 54, 10, 1

Offset: 1

Mats Granvik, May 16 2010

The first column in this array is the first column in A134540 which is the matrix inverse of A177978. The second column is the first column in A159905. The rows in this array are described by both binomial expressions and closed form polynomials.

			Table begins:
  1..1...1...1....1.....1.....1......1......1.......1.......1
  0..1...2...3....4.....5.....6......7......8.......9......10
  0..2...5...9...14....20....27.....35.....44......54......65
  0..2...7..16...30....50....77....112....156.....210.....275
  0..4..14..34...69...125...209....329....494.....714....1000
  0..2..13..43..107...226...428....749...1234....1938....2927
  0..6..27..83..209...461...923...1715...3002....5004....8007
  0..4..26.100..295...736..1632...3312...6270...11220...19162
  0..6..39.155..480..1266..2975...6399..12825...24255...43692
  0..4..38.182..641..1871..4789..11103..23807...47896...91367
  0.10..65.285.1000..3002..8007..19447..43757...92377..184755
  0..4..50.292.1209..4066.11837..30920..74139..165748..349438
  0.12..90.454.1819..6187.18563..50387.125969..293929..646645
  0..6..75.473.2166..8101.26202..75797.200479..492406.1136048
  0..8.100.636.2976.11482.38523.115915.319231..816421.1960190
  0..8.100.696.3546.14712.52548.167112.483879.1296064.3249312

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Column k=1..5 gives A063524, A000010, A007438, A117108, A117109.
Main diagonal gives A332470.
Cf. A177976, A177977.

T(n, k) = sumdiv(n, d, moebius(n/d)*binomial(d+k-2, d-1)); \\ Seiichi Manyama, Jun 12 2021

From Seiichi Manyama, Jun 12 2021: (Start)
G.f. of column k: Sum_{j>=1} mu(j) * x^j/(1 - x^j)^k.
T(n,k) = Sum_{d|n} mu(n/d) * binomial(d+k-2,d-1). (End)

A338656 a(n) = Sum_{d|n} mu(d) * binomial(d + n/d - 2, d-1).

Original entry on oeis.org

1, 0, 0, -1, 0, -4, 0, -3, -5, -8, 0, -9, 0, -12, -28, -7, 0, -8, 0, -34, -54, -20, 0, 9, -69, -24, -44, -83, 0, 0, 0, -15, -130, -32, -418, 157, 0, -36, -180, -129, 0, 0, 0, -285, -494, -44, 0, 633, -923, -24, -304, -454, 0, 1090, -2000, -1183, -378, -56, 0, 3050, 0, -60, -3002, -31, -3638, 0

Offset: 1

Seiichi Manyama, Apr 22 2021

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

Cf. A008683, A157020, A332470, A338657.

a[n_] := DivisorSum[n, MoebiusMu[#] * Binomial[# + n/# - 2, # - 1] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)

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

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

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

If p is prime, a(p) = 0.

A343565 a(n) = |{(x_1, x_2, ... , x_n) : 1 <= x_1 <= x_2 <= ... <= x_n <= n, gcd(x_1, x_2, ... , x_n, n) = 1}|.

Original entry on oeis.org

1, 2, 9, 30, 125, 428, 1715, 6270, 24255, 91367, 352715, 1345448, 5200299, 20019526, 77554749, 300295038, 1166803109, 4535971916, 17672631899, 68913247655, 269128640958, 1051984969598, 4116715363799, 16123381989000, 63205303195125, 247956558998878, 973469689288236

Offset: 1

Seiichi Manyama, Apr 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000010, A007438, A117108, A117109, A332470, A343517, A343548, A343549.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[# + n - 1, n] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)

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

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

a(n) = [x^n] Sum_{k>=1} mu(k) * x^k/(1 - x^k)^(n+1).

A345131 Number of ordered n-tuples of integers from [ 1..n ] with no global factor.

Original entry on oeis.org

1, 2, 8, 29, 118, 427, 1671, 6260, 24034, 91301, 351261, 1345434, 5191170, 20018845, 77500485, 300290041, 1166450850, 4535971707, 17670369300, 68913194733, 269114332057, 1051984590581, 4116622325140, 16123381985750, 63204699026898, 247956554702702

Offset: 1

Seiichi Manyama, Jun 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Main diagonal of A177976.

Cf. A332470.

a[n_] := Sum[DivisorSum[k, MoebiusMu[k/#] * Binomial[n + # - 2, # - 1] &], {k, 1, n}]; Array[a, 25] (* Amiram Eldar, Jun 13 2021 *)

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

a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * binomial(d+n-2, d-1).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^n.
a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 19 2021

cp's OEIS Frontend

A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

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A343553 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).

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A177975 Square array T(n,k) read by antidiagonals up. Each column is the first column in the matrix inverse of a triangular matrix that is the k-th differences of A051731 in the column direction.

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A338656 a(n) = Sum_{d|n} mu(d) * binomial(d + n/d - 2, d-1).

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A343565 a(n) = |{(x_1, x_2, ... , x_n) : 1 <= x_1 <= x_2 <= ... <= x_n <= n, gcd(x_1, x_2, ... , x_n, n) = 1}|.

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A345131 Number of ordered n-tuples of integers from [ 1..n ] with no global factor.

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