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A140256
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Triangle read by columns: Column k is A014963 aerated with groups of (k-1) zeros.
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7
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1, 2, 1, 3, 0, 1, 2, 2, 0, 1, 5, 0, 0, 0, 1, 1, 3, 2, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 1, 3, 0, 3, 0, 0, 0, 0, 0, 1, 1, 5, 0, 0, 2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,2
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COMMENTS
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If the row number n is prime, the row consists of T(n,1)=n followed by n-2 zeros and followed by T(n,n)=1.
Row products of nonzero terms in row n, equals n. - Mats Granvik, May 22 2016
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LINKS
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FORMULA
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Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s)*(1 - 1/n^(s - 1)) -Sum_{i=2..n} Ts(n, i)/(i)^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.
For n not equal to k: Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.
Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then log(n) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0. (End)
[The above sentences need a lot of work! Parentheses might help. - N. J. A. Sloane, Mar 14 2017]
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EXAMPLE
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First few rows of the triangle are:
1;
2, 1;
3, 0, 1;
2, 2, 0, 1;
5, 0, 0, 0, 1;
1, 3, 2, 0, 0, 1;
7, 0, 0, 0, 0, 0, 1;
2, 2, 0, 2, 0, 0, 0, 1;
3, 0, 3, 0, 0, 0, 0, 0, 1;
1, 5, 0, 0, 2, 0, 0, 0, 0, 1;
11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1;
...
Column 2 = (1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 1, 0, 7, ...).
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MATHEMATICA
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t[n_, k_] /; Divisible[n, k] := Exp[ MangoldtLambda[n/k] ]; t[_, _] = 0; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)
(* recurrence *)
Clear[t, s, n, k, z, nn]; z = 1; nn = 14; t[n_, k_] := t[n, k] = If[k == 1, Zeta[s]*(1 - 1/n^(s - 1)) -Sum[t[n, i]/i^(s - 1), {i, 2, n}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; A = Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]; Flatten[Exp[A]*Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Apr 09 2016, May 22 2016 *)
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PROG
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(Excel) =if(row()>=column(); if(mod(row(); column())=0; lookup(roundup(row()/column(); 0); A000027; A014963); 0); "")
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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