A117485 -id:A117485 - cp's OEIS frontend

A164679 Convolve A001399 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

1, 2, 1, 5, 4, 2, 10, 9, 7, 3, 19, 18, 16, 11, 4, 33, 32, 30, 25, 16, 5, 57

Offset: 1

Alford Arnold, Sep 05 2009

Apparently the terms can be constructed by fixing the generating function of the diagonal g_0(x) = 1/(1-x)/(1-x^2)/(1-x^3), A001399, and deriving the generating function of the i-th subdiagonal by g_i(x) = g_{i-1}(x)/(1-x^i), i>=1. - R. J. Mathar, May 17 2016

			1;
2, 1;
5, 4, 2;
10, 9, 7, 3;
19, 18, 16, 11, 4;
33, 32, 30, 25, 16, 5;
57

Cf. A000098 (first column), A164678 (a similar triangle). Diagonals are A001399, A000601, A097701, A117485, ...

A364842 Table read by antidiagonals: row n gives the Euler transform of the sequence (2,...,2,0,0,...) that contains n 2s followed by 0s.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 8, 5, 1, 2, 5, 10, 14, 6, 1, 2, 5, 10, 18, 20, 7, 1, 2, 5, 10, 20, 30, 30, 8, 1, 2, 5, 10, 20, 34, 49, 40, 9, 1, 2, 5, 10, 20, 36, 59, 74, 55, 10, 1, 2, 5, 10, 20, 36, 63, 94, 110, 70, 11, 1, 2, 5, 10, 20, 36, 65, 104, 149, 158, 91, 12

Offset: 1

Peter Kagey, Nov 09 2023

			Table begins:
  | 0 1 2  3  4  5  6   7   8   9  10
--+----------------------------------
1 | 1 2 3  4  5  6  7   8   9  10  11
2 | 1 2 5  8 14 20 30  40  55  70  91
3 | 1 2 5 10 18 30 49  74 110 158 221
4 | 1 2 5 10 20 34 59  94 149 224 334
5 | 1 2 5 10 20 36 63 104 169 264 405
6 | 1 2 5 10 20 36 65 108 179 284 445
7 | 1 2 5 10 20 36 65 110 183 294 465
8 | 1 2 5 10 20 36 65 110 185 298 475
9 | 1 2 5 10 20 36 65 110 185 300 479

Cf. A000027 (row 1), A006918 (row 2), A117485 (row 3), A117486 ( row 4), A117487 (row 5), A160647 (row 6), A000712 (main diagonal).
Analogous for initial 1s sequence A008284.
Cf. A115994.

Seed[i_, n_] := ConstantArray[2, i]~Join~ConstantArray[0, n - i];
A364842Table[n_] := Table[Seed[i, n] // EulerTransform, {i, 1, n}]
(*EulerTransform is defined in A005195*)

cp's OEIS Frontend

A164679 Convolve A001399 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

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