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.
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
Offset: 0
Examples
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;
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
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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-François Alcover, Mar 25 2017, translated from Maple *) -
Sage
# uses[DivisorTriangle from A327029] DivisorTriangle(euler_phi, stirling_number2, 10) # Peter Luschny, Aug 24 2019
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)
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