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A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals. 23
0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 20, 33, 24, 5, 0, 0, 6, 30, 72, 96, 40, 6, 0, 0, 7, 42, 135, 280, 255, 84, 7, 0, 0, 8, 56, 228, 660, 1040, 780, 140, 8, 0, 0, 9, 72, 357, 1344, 3145, 4200, 2205, 288, 9, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

FORMULA

A(n,k) = Sum_{d|n} phi(d)*k^(n/d).

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258170(n,i). - Alois P. Heinz, May 22 2015

G.f. for column k: Sum_{n>=1} phi(n)*k*x^n/(1-k*x^n) for k >= 0. - Petros Hadjicostas, Nov 06 2017

EXAMPLE

Square array A(n,k) begins:

  0, 0,  0,   0,    0,     0,     0, ...

  0, 1,  2,   3,    4,     5,     6, ...

  0, 2,  6,  12,   20,    30,    42, ...

  0, 3, 12,  33,   72,   135,   228, ...

  0, 4, 24,  96,  280,   660,  1344, ...

  0, 5, 40, 255, 1040,  3145,  7800, ...

  0, 6, 84, 780, 4200, 15810, 46956, ...

MAPLE

with(numtheory):

A:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)):

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

a[_, 0] = a[0, _] = 0; a[n_, k_] := Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Dec 06 2013 *)

CROSSREFS

Columns k=0..10 give A000004, A001477, A053635, A054610, A054611, A054612, A054613, A054614, A054615, A054616, A054617.

Rows n=0..10 give A000004, A001477, A002378, A054602, A054603, A054604, A054605, A054606, A054607, A054608, A054609.

Main diagonal gives A228640.

Cf. A000010, A258170.

Sequence in context: A014473 A226545 A271917 * A265080 A228275 A228250

Adjacent sequences:  A185648 A185649 A185650 * A185652 A185653 A185654

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 29 2013

STATUS

approved

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Last modified August 25 14:35 EDT 2019. Contains 326324 sequences. (Running on oeis4.)