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A309229 Square array read by upwards antidiagonals. See formula section for recurrence. 1
1, 2, 1, 3, 0, 1, 4, 1, 2, 1, 5, 0, 0, 0, 1, 6, 1, 1, 1, 2, 1, 7, 0, 2, 0, 3, 0, 1, 8, 1, 0, 1, 4, -2, 2, 1, 9, 0, 1, 0, 0, -3, 3, 0, 1, 10, 1, 2, 1, 1, -2, 4, 1, 2, 1, 11, 0, 0, 0, 2, 0, 5, 0, 0, 0, 1, 12, 1, 1, 1, 3, 1, 6, 1, 1, 1, 2, 1, 13, 0, 2, 0, 4, 0, 0, 0, 2, 0, 3, 0, 1, 14, 1, 0, 1, 0, -2, 1, 1, 0, -4, 4, -2, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

log(A003418(n)) = Sum_{k>=1} (T(n, k)/k - 1/k).

LINKS

Table of n, a(n) for n=1..105.

Mathematics Stack Exchange, Do these series converge to the von Mangoldt function?

FORMULA

Recurrence:

T(n, 1) = if n >= 1 then n, otherwise 0;

T(1, k) = 1;

T(n, k) = if(and(n > 1, k > 1) then If(n > k then T(n - k, k) else Sum_{i=0..n-1} T(n - 1, k - i) - Sum_{i=1..n-1} T(n, k - i)) else 0)

EXAMPLE

   1, 1, 1, 1, 1,  1, 1, 1, 1,  1,  1,  1,  1,  1, ...

   2, 0, 2, 0, 2,  0, 2, 0, 2,  0,  2,  0,  2,  0, ...

   3, 1, 0, 1, 3, -2, 3, 1, 0,  1,  3, -2,  3,  1, ...

   4, 0, 1, 0, 4, -3, 4, 0, 1,  0,  4, -3,  4,  0, ...

   5, 1, 2, 1, 0, -2, 5, 1, 2, -4,  5, -2,  5,  1, ...

   6, 0, 0, 0, 1,  0, 6, 0, 0, -5,  6,  0,  6,  0, ...

   7, 1, 1, 1, 2,  1, 0, 1, 1, -4,  7,  1,  7, -6, ...

   8, 0, 2, 0, 3,  0, 1, 0, 2, -5,  8,  0,  8, -7, ...

   9, 1, 0, 1, 4, -2, 2, 1, 0, -4,  9, -2,  9, -6, ...

  10, 0, 1, 0, 0, -3, 3, 0, 1,  0, 10, -3, 10, -7, ...

  11, 1, 2, 1, 1, -2, 4, 1, 2,  1,  0, -2, 11, -6, ...

  12, 0, 0, 0, 2,  0, 5, 0, 0,  0,  1,  0, 12, -7, ...

  13, 1, 1, 1, 3,  1, 6, 1, 1,  1,  2,  1,  0, -6, ...

  14, 0, 2, 0, 4,  0, 0, 0, 2,  0,  3,  0,  1,  0, ...

  ...

MATHEMATICA

Clear[nn, t, n, k]; nn = 14; t[n_, 1] = If[n >= 1, n, 0]; t[1, k_] = 1; t[n_, k_] := t[n, k] = If[And[n > 1, k > 1], If[n > k, t[n - k, k], Sum[t[n - 1, k - i], {i, 0, n - 1}] - Sum[t[n, k - i], {i, 1, n - 1}]], 0]; TableForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]; Flatten[Table[Table[t[n - k + 1, k], {k, 1, n}], {n, 1, nn}]]

CROSSREFS

Cf. A191898, A003418.

Sequence in context: A127094 A221642 A158906 * A143239 A158951 A126988

Adjacent sequences:  A309226 A309227 A309228 * A309230 A309231 A309232

KEYWORD

tabl,sign

AUTHOR

Mats Granvik, Aug 10 2019

STATUS

approved

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Last modified January 21 16:22 EST 2020. Contains 331114 sequences. (Running on oeis4.)