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A258170 T(n,k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i); triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. 4
0, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 8, 6, 1, 0, 5, 15, 25, 10, 1, 0, 6, 36, 91, 65, 15, 1, 0, 7, 63, 301, 350, 140, 21, 1, 0, 8, 136, 972, 1702, 1050, 266, 28, 1, 0, 9, 261, 3027, 7770, 6951, 2646, 462, 36, 1, 0, 10, 530, 9355, 34115, 42526, 22827, 5880, 750, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

FORMULA

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i).

From Petros Hadjicostas, Sep 07 2018: (Start)

Conjecture 1: T(n,k) = Stirling2(n,k) for k >= 1 and k <= n <= 2*k - 1.

Conjecture 2: T(n,k) = Stirling2(n,k) for k >= 2 and n prime >= 2.

Here, Stirling2(n,k) = A008277(n,k).

(End)

EXAMPLE

Triangle T(n,k) begins:

  0;

  0,  1;

  0,  2,   1;

  0,  3,   3,    1;

  0,  4,   8,    6,     1;

  0,  5,  15,   25,    10,     1;

  0,  6,  36,   91,    65,    15,     1;

  0,  7,  63,  301,   350,   140,    21,    1;

  0,  8, 136,  972,  1702,  1050,   266,   28,   1;

  0,  9, 261, 3027,  7770,  6951,  2646,  462,  36,  1;

  0, 10, 530, 9355, 34115, 42526, 22827, 5880, 750, 45, 1;

MAPLE

with(numtheory):

A:= proc(n, k) option remember;

      add(phi(d)*k^(n/d), d=divisors(n))

    end:

T:= (n, k)-> add((-1)^(k-i)*binomial(k, i)*A(n, i), i=0..k)/k!:

seq(seq(T(n, k), k=0..n), n=0..12);

MATHEMATICA

A[n_, k_] := A[n, k] = DivisorSum[n, EulerPhi[#]*k^(n/#)&];

T[n_, k_] := Sum[(-1)^(k-i)*Binomial[k, i]*A[n, i], {i, 0, k}]/k!; T[0, 0] = 0;

Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Mar 25 2017, translated from Maple *)

PROG

(SageMath) # Function DivisorTriangle is defined in A327029, returns T(0, 0) = 1.

DivisorTriangle(euler_phi, stirling_number2, 10) # Peter Luschny, Aug 24 2019

CROSSREFS

Columns k=0-1 give: A000004, A000027.

Row sums give A258171.

Main diagonal gives A057427.

T(2*n+1,n+1) gives A129506(n+1).

Cf. A008277, A185651, A327029.

Sequence in context: A089000 A253829 A107238 * A055830 A293109 A233530

Adjacent sequences:  A258167 A258168 A258169 * A258171 A258172 A258173

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 22 2015

STATUS

approved

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Last modified September 18 12:00 EDT 2019. Contains 327170 sequences. (Running on oeis4.)