A340524 - OEIS

A340524

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

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

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OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..91.

EXAMPLE

Triangle begins:

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.

PROG

(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

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).

Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340426.

Sequence in context: A268189 A265247 A022879 * A064984 A323226 A307197

Adjacent sequences: A340521 A340522 A340523 * A340525 A340526 A340527

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Jan 10 2021

STATUS

approved