A363628 -id:A363628 - cp's OEIS frontend

A363639 Expansion of Sum_{k>0} (1/(1 - k*x^k)^3 - 1).

3, 12, 19, 51, 36, 180, 57, 405, 352, 918, 111, 3990, 144, 5064, 6534, 15945, 222, 51462, 267, 99354, 82478, 160812, 369, 808490, 66051, 861630, 1090342, 2593614, 552, 8966414, 621, 13039761, 13831470, 22415778, 3166218, 114011229, 852, 110103540, 167426822

Offset: 1

Seiichi Manyama, Jun 13 2023

nonn

Cf. A338662, A362683, A363640.
Cf. A363642, A363643, A363644.
Cf. A363628, A363647.

a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + 2, 2] &]; Array[a, 40] (* Amiram Eldar, Jul 17 2023 *)

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

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

A366984 a(n) = Sum_{k=1..n} binomial(k+2,2) * floor(n/k).

Original entry on oeis.org

3, 12, 25, 49, 73, 120, 159, 228, 296, 392, 473, 626, 734, 899, 1069, 1291, 1465, 1757, 1970, 2312, 2614, 2977, 3280, 3803, 4178, 4670, 5144, 5759, 6227, 6993, 7524, 8307, 8993, 9803, 10529, 11630, 12374, 13373, 14311, 15559, 16465, 17867, 18860, 20273, 21579, 23016

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363628.
Cf. A006218, A366983, A366985.
Cf. A366967.

a(n) = sum(k=1, n, binomial(k+2, 2)*(n\k));

from math import isqrt
def A366984(n): return (-(s:=isqrt(n))*(s*(s*(s+7)+17)+17)+sum(((q:=n//w)+1)*(q*(q+5)+3*(w*(w+3)+4)) for w in range(1,s+1)))//6 # Chai Wah Wu, Oct 31 2023

G.f.: 1/(1-x) * Sum_{k>0} (1/(1-x^k)^3 - 1) = 1/(1-x) * Sum_{k>0} binomial(k+2,2) * x^k/(1-x^k).

A363660 a(n) = Sum_{d|n} binomial(d+n,n).

Original entry on oeis.org

2, 9, 24, 90, 258, 1043, 3440, 13419, 48850, 187836, 705444, 2725099, 10400614, 40233015, 155133856, 601820876, 2333606238, 9079958260, 35345263820, 137876637843, 538259060526, 2104292500739, 8233430727624, 32248866496625, 126410606580284

Offset: 1

Seiichi Manyama, Jun 14 2023

nonn

Cf. A007503, A116963, A363628.

Cf. A000984, A134758, A343548, A343549, A363661.

a[n_] := DivisorSum[n, Binomial[# + n, n] &]; Array[a, 25] (* Amiram Eldar, Jul 17 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+n, n));
```

a(n) = [x^n] Sum_{k>0} (1/(1 - x^k)^(n+1) - 1).

a(n) = [x^n] Sum_{k>0} binomial(k+n,n) * x^k/(1 - x^k).

A363695 Expansion of Sum_{k>0} (1/(1-x^k)^5 - 1).

Original entry on oeis.org

5, 20, 40, 90, 131, 265, 335, 585, 755, 1147, 1370, 2155, 2385, 3410, 4042, 5430, 5990, 8295, 8860, 11843, 13020, 16335, 17555, 23125, 23882, 29805, 32220, 39440, 40925, 51644, 52365, 64335, 67450, 79820, 82712, 101575, 101275, 120805, 125830, 148089, 149000, 179490

Offset: 1

Seiichi Manyama, Jun 16 2023

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

Cf. A007503, A116963, A363628, A363696.

Cf. A073570, A363605, A363608.

a[n_] := DivisorSum[n, Binomial[# + 4, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 05 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+4, 4));
```

G.f.: Sum_{k>0} binomial(k+4,4) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d+4,4).

A363696 Expansion of Sum_{k>0} (1/(1-x^k)^6 - 1).

Original entry on oeis.org

6, 27, 62, 153, 258, 545, 798, 1440, 2064, 3282, 4374, 6859, 8574, 12447, 15818, 21789, 26340, 36196, 42510, 56538, 66634, 85125, 98286, 126901, 142764, 178506, 203440, 249909, 278262, 343936, 376998, 457686, 506372, 602118, 659058, 791908, 850674, 1005129, 1094638

Offset: 1

Seiichi Manyama, Jun 16 2023

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

Cf. A007503, A116963, A363628, A363695.

Cf. A101289, A363606.

a[n_] := DivisorSum[n, Binomial[# + 5, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 05 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+5, 5));
```

G.f.: Sum_{k>0} binomial(k+5,5) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d+5,5).

cp's OEIS Frontend

A363639 Expansion of Sum_{k>0} (1/(1 - k*x^k)^3 - 1).

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A366984 a(n) = Sum_{k=1..n} binomial(k+2,2) * floor(n/k).

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A363660 a(n) = Sum_{d|n} binomial(d+n,n).

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A363695 Expansion of Sum_{k>0} (1/(1-x^k)^5 - 1).

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A363696 Expansion of Sum_{k>0} (1/(1-x^k)^6 - 1).

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