A156833 -id:A156833

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A157020

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

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

MAPLE

add( d*binomial(n/d+d-2, d-1), d=numtheory[divisors](n) ) ;

PROG

(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

CROSSREFS

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

KEYWORD

easy,nonn,look

AUTHOR

R. J. Mathar, Feb 21 2009

STATUS

approved

A156834

A156348 * A000010

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A156348, A000010, A156833, A157020, A343553.

Equals row sums of triangle A157030. [Gary W. Adamson, Feb 21 2009]

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson, Feb 16 2009

EXTENSIONS

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

STATUS

approved

A145352

Triangle read by rows, A054525 * A157497

+10
0

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

Table of n, a(n) for n=1..78.

FORMULA

Inverse Mobius transform of triangle A157497.

EXAMPLE

First few rows of the triangle =

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

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson & Mats Granvik, Mar 01 2009

STATUS

approved

A156839

Triangle read by rows, A054525 * A156348 * A000012

+10
0

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,7

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,...).

LINKS

Table of n, a(n) for n=1..79.

FORMULA

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

EXAMPLE

First few rows of the triangle =

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

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson and Mats Granvik, Feb 16 2009

STATUS

approved

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