cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 3, 0, 2, 1, 2, 0, 2, 2, 2, 4, 0, 3, 2, 4, 2, 2, 0, 2, 3, 4, 4, 4, 4, 0, 4, 2, 6, 4, 8, 4, 3, 0, 2, 4, 4, 6, 8, 8, 7, 4, 0, 4, 2, 8, 4, 12, 8, 14, 8, 2, 0, 3, 4, 4, 8, 8, 12, 14, 16, 12, 6, 0, 4, 3, 8, 4, 16, 8, 21, 16, 24, 14, 2, 0, 2, 4, 6, 8, 8, 16, 14, 24, 24, 28, 21
Offset: 1

Views

Author

Omar E. Pol, Jan 10 2021

Keywords

Comments

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

Examples

			Triangle begins:
1;
2, 0;
2, 0, 1;
3, 0, 2, 1;
2, 0, 2, 2, 2;
4, 0, 3, 2, 4, 2;
2, 0, 2, 3, 4, 4,  4;
4, 0, 4, 2, 6, 4,  8,  4;
3, 0, 2, 4, 4, 6,  8,  8,  7;
4, 0, 4, 2, 8, 4, 12,  8, 14,  8;
2, 0, 3, 4, 4, 8,  8, 12, 14, 16, 12;
6, 0, 4, 3, 8, 4, 16,  8, 21, 16, 24, 14;
2, 0, 2, 4, 6, 8,  8, 16, 14, 24, 24, 28, 21;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *   4   =   4
2      0   *   2   =   0
3      1   *   3   =   3
4      1   *   2   =   2
5      2   *   2   =   4
6      2   *   1   =   2
.           A000005
--------------------------
The sum of row 6 is 4 + 0 + 3 + 2 + 4 + 2 = 15, equaling A138137(6) = 15.
		

Crossrefs

Row sums give A138137 (conjectured).
Columns 1, 3 and 4 are A000005.
Column 2 gives A000004.
Columns 5 and 6 give A062011.
Columns 7 and 8 give A145154, n >= 1.
Leading diagonal gives A002865.
Cf. A339304 (irregular or expanded version).

Programs

  • PARI
    f(n) = if (n==0, 1, numbpart(n) - numbpart(n-1)); \\ A002865
    T(n, k) = numdiv(n-k+1) * f(k-1); \\ Michel Marcus, Jan 13 2021

A145229 Coefficients in expansion of E'_1(q).

Original entry on oeis.org

4, 16, 24, 48, 40, 96, 56, 128, 108, 160, 88, 288, 104, 224, 240, 320, 136, 432, 152, 480, 336, 352, 184, 768, 300, 416, 432, 672, 232, 960, 248, 768, 528, 544, 560, 1296, 296, 608, 624, 1280, 328, 1344, 344, 1056, 1080, 736, 376, 1920, 588, 1200, 816, 1248, 424, 1728
Offset: 0

Views

Author

N. J. A. Sloane, Feb 28 2009

Keywords

Comments

Derivative of E_1(q): see A145154.
Equals 4 times A038040.
Showing 1-2 of 2 results.