A056045 -id:A056045 - cp's OEIS frontend

A082905 Modified Pascal-triangle, read by rows. All C(n,j) binomial coefficients are replaced by C(n/g, j/g), where g = gcd(n,j).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 2, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 3, 2, 3, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 4, 56, 2, 56, 4, 8, 1, 1, 9, 36, 3, 126, 126, 3, 36, 9, 1, 1, 10, 5, 120, 10, 2, 10, 120, 5, 10, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 1, 12, 6, 4, 3, 792, 2, 792, 3, 4, 6, 12, 1

Offset: 0

Labos Elemer, Apr 23 2003

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  4,  2,   4,   1;
  1,  5, 10,  10,   5,   1;
  1,  6,  3,   2,   3,   6,  1;
  1,  7, 21,  35,  35,  21,  7,   1;
  1,  8,  4,  56,   2,  56,  4,   8, 1;
  1,  9, 36,   3, 126, 126,  3,  36, 9,  1;
  1, 10,  5, 120,  10,   2, 10, 120, 5, 10, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000005, A007318, A056045.

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return Binomial(n/Gcd(n,k), k/Gcd(n,k));
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Aug 30 2019

Flatten[Table[Table[Binomial[n/GCD[n, j], j/GCD[n, j]], {j, 0, n}], {n, 1, 32}], 1]

T(n,k) = my(g=gcd(n,k)); if (!g, g=1); binomial(n/g, k/g);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", "))); \\ Michel Marcus, Aug 30 2019

def T(n,k):
    if k==0 or k==n: return 1
    else: return binomial(n/gcd(n,k), k/gcd(n,k))
[[T(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 30 2019

More terms from Michel Marcus, Aug 30 2019

A308812 a(n) = Sum_{k=1..n} binomial(n,k) * floor(n/k).

Original entry on oeis.org

1, 5, 13, 33, 61, 143, 246, 521, 985, 1995, 3499, 7923, 14028, 28642, 55603, 115369, 210665, 455399, 838338, 1755983, 3383652, 6974159, 13034492, 28011611, 52475486, 108821068, 210050941, 436273458, 824191369, 1744975533, 3301974301, 6867107913, 13250454241

Offset: 1

Ilya Gutkovskiy, Aug 22 2019

nonn

Robert Israel, Table of n, a(n) for n = 1..3318

Cf. A056045.

f:= proc(n) local k; add(binomial(n,k)*floor(n/k),k=1..n) end proc:
map(f, [$1..100]); # Robert Israel, Aug 23 2019

Table[Sum[Binomial[n, k] Floor[n/k] , {k, 1, n}], {n, 1, 33}]
Table[SeriesCoefficient[1/(1 - x) Sum[Binomial[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 33}]
Table[Sum[Sum[Binomial[n, d], {d, Divisors[k]}], {k, 1, n}], {n, 1, 33}]

a(n) = [x^n] (1/(1 - x)) * Sum_{k=1..n} binomial(n,k) * x^k/(1 - x^k).
a(n) = Sum_{k=1..n} Sum_{d|k} binomial(n,d).
a(n) ~ 3 * 2^(n-1). - Vaclav Kotesovec, May 28 2021

A327124 Expansion of Sum_{k>=1} ((1 - (-x)^k)^k - 1).

Original entry on oeis.org

1, -2, 3, -3, 5, -3, 7, -2, 10, 0, 11, -1, 13, 7, 25, 13, 17, -2, 19, 30, 56, 33, 23, 1, 26, 52, 111, 98, 29, -51, 31, 158, 198, 102, 56, 24, 37, 133, 325, 304, 41, -189, 43, 517, 626, 207, 47, 191, 50, -2, 731, 988, 53, -435, 517, 1315, 1026, 348, 59, 18

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

sign

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

Cf. A056045, A112329, A318636, A327238.

nmax = 60; CoefficientList[Series[Sum[((1 - (-x)^k)^k - 1), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(n - #) Binomial[n/#, #] &], {n, 1, 60}]

a(n)={sumdiv(n, d, (-1)^(n-d) * binomial(n/d,d))} \\ Andrew Howroyd, Sep 14 2019

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

a(p) = p, where p is odd prime.

A330017 a(1) = 1; a(n+1) = Sum_{d|n} binomial(n,d) * a(d).

Original entry on oeis.org

1, 1, 3, 6, 16, 21, 102, 109, 565, 826, 4913, 4924, 28036, 28049, 378218, 427646, 1841532, 1841549, 29704312, 29704331, 182590131, 194454702, 3660242371, 3660242394, 17058569521, 17059419626, 308650641577, 311298706504, 1436650115240, 1436650115269

Offset: 1

Ilya Gutkovskiy, Nov 27 2019

nonn

Cf. A003238, A056045.

a[n_] := a[n] = Sum[Binomial[n - 1, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 30}]

A105863 a(n) = n * Sum_{d|n} (binomial(n,d) / gcd(n,d)).

Original entry on oeis.org

1, 6, 12, 44, 30, 252, 56, 856, 846, 3080, 132, 20616, 182, 49532, 52110, 237232, 306, 1227096, 380, 4106320, 2470272, 15525092, 552, 86092176, 1328900, 270424752, 126624006, 1157002616, 870, 5577100260, 992, 19572325728, 6386892930

Offset: 1

Robert G. Wilson v, Apr 23 2005

nonn

Cf. A056045, A105861, A105862.

f[n_] := Block[{d = Divisors[n]}, n*Plus @@ (Binomial[n, d])]; Table[ f[n], {n, 34}]

a(n) = n * A056045(n) = n * Sum_{d|n} (binomial(n, d) / gcd(n, d)).

A308943 a(n) = Product_{d|n} binomial(n,d).

Original entry on oeis.org

1, 2, 3, 24, 5, 1800, 7, 15680, 756, 113400, 11, 79693891200, 13, 4372368, 20495475, 44972928000, 17, 2028339316523520, 19, 52737518268864000, 3247700400, 3585005424, 23, 38135556819759802035135799296, 1328250, 87885070000, 370142004375, 10293527616645873600000, 29

Offset: 1

Ilya Gutkovskiy, Jul 01 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..419

Cf. A001142, A008578 (fixed points), A056045 (similar, with Sum), A098710, A135396.

Cf. A000010 (comments on product formulas).

Table[Product[Binomial[n, d], {d, Divisors[n]}], {n, 1, 29}]

a(n) = my(p=1); fordiv(n, d, p *= binomial(n, d)); p; \\ Michel Marcus, Jul 02 2019

from math import prod, comb
from sympy import divisors
def A308943(n): return prod(comb(n,d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 22 2024

a(n) = Product_{k=1..n} binomial(n,gcd(n,k))^(1/phi(n/gcd(n,k))) = Product_{k=1..n} binomial(n,n/gcd(n,k))^(1/phi(n/gcd(n,k))) where phi = A000010. - Richard L. Ollerton, Nov 08 2021

A345136 a(1) = 1; a(n) = Sum_{d|n, d < n} binomial(n,d) * a(d).

Original entry on oeis.org

1, 2, 3, 16, 5, 96, 7, 1184, 261, 1360, 11, 97428, 13, 24220, 16395, 15267456, 17, 14474736, 19, 251423600, 817971, 7760236, 23, 264344406312, 265675, 135208476, 1223270127, 971632668664, 29, 2584070688810, 31, 9176980861031424, 2128920321, 39671306896, 48694835

Offset: 1

Ilya Gutkovskiy, Jun 09 2021

nonn

Cf. A008578 (fixed points), A056045, A074206, A330017.

a[1] = 1; a[n_] := a[n] = Sum[If[d < n, Binomial[n, d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 35}]

A056189 a(n) = 2^n - A056188(n).

Original entry on oeis.org

1, 2, 2, 8, 2, 52, 2, 128, 170, 764, 2, 2488, 2, 11624, 16928, 32768, 2, 181324, 2, 555296, 931610, 2802584, 2, 11007664, 6643782, 43955032, 44739242, 136585808, 2, 720895864, 2, 2147483648, 3250384970, 10923540812, 11517062218

Offset: 1

Labos Elemer, Aug 02 2000

nonn

For n > 1, a(n) is the number of binary words of length n such that the numbers of 0's and 1's are not coprime. - Bartlomiej Pawlik, Sep 03 2023

			For n=6, a(6)=52 because the sum of coefficients is restricted only to k=1,5 so a(6)=64-6-6.

Cf. A056045, A056188.

a(n) = if (n==1, 1, 2^n - sum(k=0, n, if (gcd(n,k) == 1, binomial(n,k)))); \\ Michel Marcus, Mar 22 2020

a(n) = 2^n-Sum{binomial[n, k]; k>0, GCD[n, k]=1}, for n>1.

a(n) = 2 for primes.

A056190 a(n) = Sum_{d|n and gcd(d, n/d)=1} binomial(n,d).

Original entry on oeis.org

1, 3, 4, 5, 6, 42, 8, 9, 10, 308, 12, 728, 14, 3538, 3474, 17, 18, 48792, 20, 20370, 117632, 705686, 24, 737520, 26, 10400952, 28, 1204544, 30, 185903342, 32, 33, 193542210, 2333606816, 7049188, 94202222, 38, 35345264542, 8122434623

Offset: 1

Labos Elemer, Aug 02 2000

nonn

			n=100 has 9 divisors of which {1,4,25,100} are unitary, so a(100) = 100 + 3921225 + 242519269720337121015504 + 1.

Alois P. Heinz, Table of n, a(n) for n = 1..3000

Cf. A056045.

a:= n-> add(`if`(igcd(d, n/d)=1, binomial(n, d), 0),
                      d=numtheory[divisors](n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 25 2019

a[n_] := Total[Binomial[n, Select[Divisors[n], CoprimeQ[#, n/#] &]]]; Array[a, 40] (* Amiram Eldar, Jul 28 2024 *)

a(n) = sumdiv(n, d, if (gcd(d, n/d)==1, binomial(n, d))); \\ Michel Marcus, Aug 25 2019

a(n) = A056045(n) for squarefree n, when all divisors are unitary.

A082906 Sum of terms in n-th row of modified Pascal's triangle displayed in A082905.

Original entry on oeis.org

1, 2, 4, 8, 12, 32, 22, 128, 140, 350, 294, 2048, 1638, 8192, 4890, 15878, 32908, 131072, 81184, 524288, 493582, 1165676, 1393770, 8388608, 5771318, 26910682, 23162026, 89478836, 131854546, 536870912, 352862112, 2147483648, 2147516556

Offset: 0

Labos Elemer, Apr 23 2003

nonn

In A082905, all binomial coefficients C(n,j) are replaced by C(n/g, j/g), where g=gcd(n,j); a(n) = Sum_{j=0..n-1} C(n/g, j/g).

			a(0)=1; a(12) = 1 + 12 + 6 + 4 + 3 + 792 + 2 + 792 + 3 + 4 + 6 + 12 + 1 = 1638.

Cf. A000005, A007318, A056045, A082905.

Table[Apply[Plus, Table[Binomial[n/GCD[n, j], j/GCD[n, j]], {j, 0, n}]], {n, 0, 32}]

cp's OEIS Frontend

A082905 Modified Pascal-triangle, read by rows. All C(n,j) binomial coefficients are replaced by C(n/g, j/g), where g = gcd(n,j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Sage

Extensions

A308812 a(n) = Sum_{k=1..n} binomial(n,k) * floor(n/k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A327124 Expansion of Sum_{k>=1} ((1 - (-x)^k)^k - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A330017 a(1) = 1; a(n+1) = Sum_{d|n} binomial(n,d) * a(d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A105863 a(n) = n * Sum_{d|n} (binomial(n,d) / gcd(n,d)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A308943 a(n) = Product_{d|n} binomial(n,d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A345136 a(1) = 1; a(n) = Sum_{d|n, d < n} binomial(n,d) * a(d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A056189 a(n) = 2^n - A056188(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A056190 a(n) = Sum_{d|n and gcd(d, n/d)=1} binomial(n,d).

Views

Author