cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A360759 a(n) = Sum_{d|n} d^(d+n/d) * binomial(d,n/d).

Original entry on oeis.org

1, 16, 243, 4112, 78125, 1680345, 40353607, 1073766400, 31381060338, 1000000781250, 34522712143931, 1283918489808640, 51185893014090757, 2177953338656796883, 98526125335697265625, 4722366482899710050304, 239072435685151324847153
Offset: 1

Views

Author

Seiichi Manyama, Feb 19 2023

Keywords

Links

Crossrefs

Cf. A318636, A327238, A338685, A338693, A338694.
Cf. A360712.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d)*binomial(d, n/d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^k*((1+k*x^k)^k-1)))

Formula

G.f.: Sum_{k>0} k^k * ( (1 + k*x^k)^k - 1 ).
If p is prime, a(p) = p^(p+2).

A366974 Expansion of Sum_{k >=1} x^(2*k)/(1-x^k)^(k+1).

Original entry on oeis.org

0, 1, 2, 4, 4, 9, 6, 14, 12, 20, 10, 42, 12, 35, 40, 59, 16, 96, 18, 121, 84, 77, 22, 281, 80, 104, 156, 281, 28, 521, 30, 407, 264, 170, 406, 1083, 36, 209, 416, 1418, 40, 1514, 42, 1068, 1632, 299, 46, 3532, 840, 1923, 884, 1847, 52, 3824, 2420, 5377, 1216, 464, 58
Offset: 1

Views

Author

Seiichi Manyama, Oct 30 2023

Keywords

Crossrefs

Cf. A081543, A366975.
Cf. A318636.

Programs

  • PARI
    a(n) = sumdiv(n, d, binomial(d+n/d-2, d));

Formula

a(n) = Sum_{d|n} binomial(d+n/d-2,d).
If p is prime, a(p) = p - 1.

A366975 Expansion of Sum_{k >=1} x^(3*k)/(1-x^k)^(k+1).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 5, 9, 8, 14, 9, 25, 11, 27, 24, 40, 15, 65, 17, 75, 55, 65, 21, 176, 44, 90, 110, 182, 27, 324, 29, 270, 197, 152, 195, 695, 35, 189, 324, 847, 39, 925, 41, 759, 1016, 275, 45, 2215, 377, 1182, 730, 1365, 51, 2338, 1418, 3072, 1025, 434, 57, 7536, 59
Offset: 1

Views

Author

Seiichi Manyama, Oct 30 2023

Keywords

Crossrefs

Cf. A081543, A366974.
Cf. A318636.

Programs

  • PARI
    a(n) = if(n<2, 0, sumdiv(n, d, binomial(d+n/d-3, d)));

Formula

a(n) = Sum_{d|n} binomial(d+n/d-3,d) for n > 1.
If p is prime, a(p) = p - 2.

A378579 G.f. A(x) satisfies x = Sum_{n>=1} ((1 + A(x)^n)^n - 1).

Original entry on oeis.org

1, -2, 5, -15, 54, -226, 1041, -5045, 25090, -126674, 646764, -3335207, 17359589, -91138625, 482237135, -2569446532, 13774698084, -74245779493, 402105384051, -2187066640025, 11941274232967, -65425584835537, 359598131529024, -1982178299221646, 10955208670488609, -60696056311093958, 337040131916813474
Offset: 1

Views

Author

Paul D. Hanna, Jan 08 2025

Keywords

Examples

			G.f.: A(x) = x - 2*x^2 + 5*x^3 - 15*x^4 + 54*x^5 - 226*x^6 + 1041*x^7 - 5045*x^8 + 25090*x^9 - 126674*x^10 + 646764*x^11 - 3335207*x^12 + ...
where
x = (1 + A(x))-1 + (1 + A(x)^2)^2-1 + (1 + A(x)^3)^3-1 + (1 + A(x)^4)^4-1 + ...
The expansions of (1 + A(x)^n)^n - 1 begin:
n=1: x - 2*x^2 + 5*x^3 - 15*x^4 + 54*x^5 - 226*x^6 + 1041*x^7 - 5045*x^8 + ...
n=2: 2*x^2 - 8*x^3 + 29*x^4 - 108*x^5 + 430*x^6 - 1848*x^7 + 8484*x^8 + ...
n=3: 3*x^3 - 18*x^4 + 81*x^5 - 336*x^6 + 1395*x^7 - 6048*x^8 + ...
n=4: 4*x^4 - 32*x^5 + 176*x^6 - 848*x^7 + 3934*x^8 - 18416*x^9 + ...
n=5: 5*x^5 - 50*x^6 + 325*x^7 - 1775*x^8 + 9000*x^9 + ...
n=6: 6*x^6 - 72*x^7 + 540*x^8 - 3300*x^9 + 18234*x^10 + ...
n=7: 7*x^7 - 98*x^8 + 833*x^9 - 5635*x^10 + 33761*x^11 + ...
n=8: 8*x^8 - 128*x^9 + 1216*x^10 - 9024*x^11 + 58336*x^12 + ...
...
the sum of which equals x.
SPECIFIC VALUES.
A(t) = 1/8 at t = 0.16352126551257248889045664875683784263524590236453...
  where t = Sum_{n>=1} ((1 + 1/8^n)^n - 1),
  also, t = Sum_{n>=1} (1/8)^(n^2) / (1 - 1/8^n)^(n+1).
A(t) = 1/9 at t = 0.14078320572038685935740333771629838603314392626246...
  where t = Sum_{n>=1} ((1 + 1/9^n)^n - 1),
  also, t = Sum_{n>=1} (1/9)^(n^2) / (1 - 1/9^n)^(n+1).
A(t) = 1/10 at t = 0.12355985214267974666409476695653610216564400778886...
  where t = Sum_{n>=1} ((1 + 1/10^n)^n - 1).
A(t) = -1/4 at t = -0.15526284433046589758223569590356891892154738705096...
A(t) = -1/5 at t = -0.13708093574671812870578995929148440226274633630611...
A(1/6) = 0.12685609485901293251324636636937755144064758593774...
  where 1/6 = Sum_{n>=1} ((1 + A(1/6)^n)^n - 1).
A(1/7) = 0.11241354571385669088090100601380487815275189296537...
A(1/8) = 0.10094983523585092678474357142194212014408583724977...
A(1/9) = 0.09162346270443389626958872306814641680247795571686...
A(-1/6) = -0.29924902046763454720023815313494776169729752567409...
A(-1/7) = -0.21417128882821263382592721321354392301151580309678...
A(-1/8) = -0.17350785733170913051439226143409362909355792177797...

Links

Crossrefs

Cf. A318636.

Programs

  • PARI
    {a(n) = my(A = serreverse( sum(m=1,n, (1 + x^m +x*O(x^n))^m - 1) ));
polcoef(A,n)}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n>=1} ((1 + A(x)^n)^n - 1).
(2) x = Sum_{n>=1} A(x)^(n^2) / (1 - A(x)^n)^(n+1), from formula by Seiichi Manyama in A318636.
(3) x = Sum_{n>=1} A(x)^n * Sum_{d|n} binomial(n/d,d), from formula by Ridouane Oudra in A318636.
(4) A(x) = Series_Reversion(G(x)), where G(x) = Sum_{n>=1} ((1 + x^n)^n - 1) is the g.f. of A318636.

A338695 a(n) = Sum_{d|n} 2^(d-1) * binomial(d, n/d).

Original entry on oeis.org

1, 4, 12, 34, 80, 204, 448, 1072, 2308, 5280, 11264, 25088, 53248, 116032, 245920, 527880, 1114112, 2369152, 4980736, 10508880, 22022336, 46193664, 96468992, 201469408, 419430416, 872734720, 1811960832, 3758844096, 7784628224, 16107909312, 33285996544, 68723417856, 141734089728
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A034729, A318636, A318637, A338694.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 2^(# - 1) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, 2^(d-1)*binomial(d, n/d));
  • PARI
    N=40; x='x+O('x^N); Vec(sum(k=1, N, (2+2*x^k)^k-2^k)/2)

Formula

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

A345179 a(1) = 1; a(n) = Sum_{d|n, d < n} binomial(n/d,d) * a(d).

Original entry on oeis.org

1, 2, 3, 6, 5, 12, 7, 20, 12, 30, 11, 54, 13, 56, 45, 78, 17, 150, 19, 140, 126, 132, 23, 414, 30, 182, 279, 420, 29, 630, 31, 692, 528, 306, 140, 1770, 37, 380, 897, 1960, 41, 1638, 43, 2486, 2040, 552, 47, 5586, 56, 1910, 2091, 4992, 53, 4212, 2365, 6874, 2964, 870, 59, 19020
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 10 2021

Keywords

Crossrefs

Cf. A008578 (fixed points), A074206, A318636, A330017, A345136.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[If[d < n, Binomial[n/d, d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]
Previous Showing 11-16 of 16 results.

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