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A221529 Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k). 6
1, 1, 3, 2, 3, 4, 3, 6, 4, 7, 5, 9, 8, 7, 6, 7, 15, 12, 14, 6, 12, 11, 21, 20, 21, 12, 12, 8, 15, 33, 28, 35, 18, 24, 8, 15, 22, 45, 44, 49, 30, 36, 16, 15, 13, 30, 66, 60, 77, 42, 60, 24, 30, 13, 18, 42, 90, 88, 105, 66, 84, 40, 45, 26, 18, 12, 56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums give A066186.

Column 1 is A000041.

Leading diagonals 1-2: A000203, A000203.

T(n,k) is the number of partitions of n that contain k as a part multiplied by the sum of divisors of k.

It appears that T(n,k) is also the number of appearances of k in the last k section of the set of partitions of n multiplied by the sum of divisors of k.

T(n,k) is also the total number of parts in all partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).

Since A000203(k) has a symmetric representation then both T(n,k) and the partial sums of row n can be represented by symmetric polycubes - for more information see A237593 and A237270. For another version see A245099. - Omar E. Pol, Jul 15 2014

LINKS

Table of n, a(n) for n=1..78.

FORMULA

T(n,k) = sigma(k)*p(n-k) = A000203(k)*A027293(n,k).

EXAMPLE

For n = 6:

-------------------------

k   A000203        T(6,k)

1      1  *  7   =    7

2      3  *  5   =   15

3      4  *  3   =   12

4      7  *  2   =   14

5      6  *  1   =    6

6     12  *  1   =   12

.         A000041

-------------------------

So row 6 is [7, 15, 12, 14, 6, 12]. Note that the sum of row 6 is 7+15+12+14+6+12 = 66 equals A066186(6) = 6*p(6) = 6*11 = 66.

Triangle begins:

1;

1,    3;

2,    3,   4;

3,    6,   4,   7;

5,    9,   8,   7,  6;

7,   15,  12,  14,  6,  12;

11,  21,  20,  21, 12,  12,  8;

15,  33,  28,  35, 18,  24,  8, 15;

22,  45,  44,  49, 30,  36, 16, 15, 13;

30,  66,  60,  77, 42,  60, 24, 30, 13, 18;

42,  90,  88, 105, 66,  84, 40, 45, 26, 18, 12;

56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28;

PROG

(PARI) T(n, k)=sigma(k)*numbpart(n-k) \\ Charles R Greathouse IV, Feb 19 2013

CROSSREFS

Cf. A000041, A000203, A027293, A066186, A135010, A138137, A182703, A221530, A245095, A245099.

Sequence in context: A233386 A093407 A147658 * A105161 A275769 A094365

Adjacent sequences:  A221526 A221527 A221528 * A221530 A221531 A221532

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Jan 20 2013

STATUS

approved

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Last modified February 27 10:31 EST 2020. Contains 332304 sequences. (Running on oeis4.)