A120111
Bi-diagonal inverse matrix of A120108.
Original entry on oeis.org
1, -2, 1, 0, -3, 1, 0, 0, -2, 1, 0, 0, 0, -5, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -7, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1
Offset: 0
Triangle begins
1;
-2, 1;
0, -3, 1;
0, 0, -2, 1;
0, 0, 0, -5, 1;
0, 0, 0, 0, -1, 1;
0, 0, 0, 0, 0, -7, 1;
0, 0, 0, 0, 0, 0, -2, 1;
0, 0, 0, 0, 0, 0, 0, -3, 1;
0, 0, 0, 0, 0, 0, 0, 0, -1, 1;
0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1;
-
A014963:= func< n | Lcm([1..n])/Lcm([1..n-1]) >;
A120111:= func< n,k | k eq n select 1 else k eq n-1 select -A014963(n+1) else 0 >;
[A120111(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, May 05 2023
-
T[n_, k_] := Switch[k, n, 1, n-1, -Exp[MangoldtLambda[n+1]], _, 0];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* Jean-François Alcover, Mar 01 2021 *)
(* Second program *)
A014963[n_]:= LCM@@Range[n]/(LCM@@Range[n-1]);
A120111[n_, k_]:= If[k==n, 1, If[k==n-1, -A014963[n+1], 0]];
Table[A120111[n,k], {n,0,20}, {k,0,n}]//Flatten (* G. C. Greubel, May 05 2023 *)
-
def A014963(n): return lcm(range(1,n+1))/lcm(range(1,n))
def A120111(n,k):
if (kA014963(n+1)
else: return 1
flatten([[A120111(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, May 05 2023
A120109
Row sums of number triangle A120108.
Original entry on oeis.org
1, 3, 10, 21, 106, 107, 750, 1501, 4504, 4505, 49556, 49557, 644242, 644243, 644244, 1288489, 21904314, 21904315, 416181986, 416181987, 416181988, 416181989, 9572185748, 9572185749, 47860928746, 47860928747, 143582786242
Offset: 0
-
List([0..30],n->Sum([0..n],k->Lcm(List([1..n+1],i->i))/Lcm(List([1..k+1],i->i)))); # Muniru A Asiru, Mar 03 2019
-
A120108:= func< n,k | Lcm([1..n+1])/Lcm([1..k+1]) >;
[(&+[A120108(n,k): k in [0..n]]): n in [0..50]]; // G. C. Greubel, May 04 2023
-
A120108[n_, k_]:= LCM@@Range[n+1]/(LCM@@Range[k+1]);
A120109[n_]:= Sum[A120108[n, k], {k,0,n}];
Table[A120109[n], {n,0,50}] (* G. C. Greubel, May 04 2023 *)
-
a(n) = lcm([1..n+1])*sum(k=0, n, 1/lcm([1..k+1])); \\ Michel Marcus, Mar 04 2019
-
def f(n): return lcm(range(1,n+2))
def A120109(n):
return sum(f(n)//f(k) for k in range(n+1))
[A120109(n) for n in range(51)] # G. C. Greubel, May 04 2023
A120110
Diagonal sums of number triangle A120108.
Original entry on oeis.org
1, 2, 7, 15, 67, 92, 461, 1065, 3016, 3956, 29478, 42231, 379107, 547556, 603421, 991923, 12709228, 18540622, 241033695, 352271227, 389226278, 407797820, 5532937710, 8097345425, 30368363481, 41503874738, 98701094676, 127342427241
Offset: 0
-
List([0..30],n->Sum([0..Int(n/2)],k->Lcm(List([1..n-k+1],i->i))/Lcm(List([1..k+1],i->i)))); # Muniru A Asiru, Mar 04 2019
-
A120108:= func< n,k | Lcm([1..n+1])/Lcm([1..k+1]) >;
[(&+[A120108(n-k,k): k in [0..Floor(n/2)]]): n in [0..50]]; // G. C. Greubel, May 04 2023
-
A120108[n_, k_]:= LCM@@Range[n+1]/(LCM@@Range[k+1]);
A120110[n_]:= Sum[A120108[n-k,k], {k,0,n/2}];
Table[A120110[n], {n,0,50}] (* G. C. Greubel, May 04 2023 *)
-
a(n) = sum(k=0, n\2, lcm([1..n-k+1])/lcm([1..k+1])); \\ Michel Marcus, Mar 04 2019
-
def f(n): return lcm(range(1,n+2))
def A120110(n):
return sum(f(n-k)//f(k) for k in range((n//2)+1))
[A120110(n) for n in range(51)] # G. C. Greubel, May 04 2023
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