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.

Showing 1-4 of 4 results.

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

Views

Author

R. J. Mathar, Feb 21 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Maple
    add( d*binomial(n/d+d-2,d-1),d=numtheory[divisors](n) ) ;
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, k*x^k/(1-x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

Formula

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

A156834 A156348 * A000010.

Original entry on oeis.org

1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, 63, 13, 56, 99, 89, 17, 154, 19, 269, 237, 132, 23, 509, 301, 182, 379, 783, 29, 1230, 31, 881, 813, 306, 2125, 2431, 37, 380, 1299, 4157, 41, 4822, 43, 3695, 6175, 552, 47, 8529, 5587, 6266, 2787
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2009

Keywords

Comments

Conjecture: for n>1, a(n) = n iff n is prime. Companion to A156833.

Examples

			a(4) = 5 = (1, 2, 0, 1) dot (1, 1, 2, 2) = (1 + 2 + 0 + 2), where row 4 of A156348 = (1, 2, 0, 1) and (1, 1, 2, 2) = the first 4 terms of Euler's phi function.

Links

Crossrefs

Cf. A156348, A000010, A156833, A157020, A343553.
Equals row sums of triangle A157030. [Gary W. Adamson, Feb 21 2009]

Programs

  • Maple
    A156834 := proc(n)
        add(A156348(n,k)*numtheory[phi](k),k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013
  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 2, #-1] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-2, d-1)); \\ Seiichi Manyama, Apr 22 2021
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*(x/(1-x^k))^k)) \\ Seiichi Manyama, Apr 22 2021

Formula

Equals A156348 * A054525 * [1, 2, 3,...]; where A054525 = the inverse Mobius transform.
a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-2, d-1). - Seiichi Manyama, Apr 22 2021
G.f.: Sum_{k >= 1} phi(k) * (x/(1 - x^k))^k. - Seiichi Manyama, Apr 22 2021

Extensions

Extended beyond a(14) by R. J. Mathar, Mar 03 2013

A145352 Triangle read by rows, A054525 * A157497.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 2, 0, 4, 0, 0, 0, 0, 5, 0, 4, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 7, 0, 4, 0, 12, 0, 0, 0, 8, 0, 0, 15, 0, 0, 0, 0, 0, 9, 0, 8, 0, 0, 20, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 4, 21, 36, 0, 30, 0, 0, 0, 0, 0, 12
Offset: 1

Views

Author

Gary W. Adamson & Mats Granvik, Mar 01 2009

Keywords

Comments

Row sums = A156833: (1, 2, 3, 6, 5, 16, 7,...)

Examples

			First few rows of the triangle =
1;
0, 2;
0, 0, 3;
0, 2, 0, 4;
0, 0, 0, 0, 5;
0, 4, 6, 0, 0, 6;
0, 0, 0, 0, 0, 0, 7;
0, 4, 0, 12, 0, 0, 0, 8;
0, 0, 15, 0, 0, 0, 0, 0, 9;
0, 8, 0, 0, 20, 0, 0, 0, 0, 10;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
0, 4, 21, 36, 0, 30, 0, 0, 0, 0, 0, 12;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
0, 12, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 14;
...

Crossrefs

Cf. A157497, A054525, A156833

Formula

Inverse Mobius transform of triangle A157497.

A156839 Triangle read by rows, A054525 * A156348 * A000012.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 4, 4, 1, 1, 1, 1, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 9, 9, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 24, 22, 15, 6, 6, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gary W. Adamson and Mats Granvik, Feb 16 2009

Keywords

Comments

For rows >1, n-th row = all 1's iff n is prime.
Row sums = A156833: (1, 2, 3, 6, 5, 16, 7, 24, 24, 38, 11,...).

Examples

			First few rows of the triangle =
1;
1, 1;
1, 1, 1;
2, 2, 1, 1;
1, 1, 1, 1, 1;
5, 5, 3, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1;
6, 6, 4, 4, 1, 1, 1, 1;
6, 6, 6, 1, 1, 1, 1, 1, 1;
9, 9, 5, 5, 5, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
24, 24, 22, 15, 6, 6, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
13, 13, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1;
...

Crossrefs

Cf. A156348, A054525, A156833

Formula

Triangle read by rows, A054525 * A156348 * A000012 A054525 = the inverse Mobius transform, A000012 = an infinite lower triangular matrix with all 1's.
Showing 1-4 of 4 results.

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