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A174739 Triangle read by rows, a partition number generator; A145006 * the diagonalized variant of A000041, (A174712). 2
1, 1, 1, 0, 1, 2, 0, 0, 2, 3, -1, 0, 0, 3, 5, 0, -1, 0, 0, 5, 7, -1, 0, -2, 0, 0, 7, 11, 0, -1, 0, -3, 0, 0, 11, 15, 0, 0, -2, 0, -5, 0, 15, 22, 0, 0, 0, -3, 0, -7, 0, 0, 22, 30, 0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42, 1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56, 0, 1, 0, 0, 0, 0, -11, 0, -22, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Row sums = the partition numbers, A000041 starting with offset 1.

The triangle demonstrates an equivalency to Euler's pentagonal recurrence

relation, such that sum of n-th row terms = rightmost term of next row, a

partition number.

Contribution from Gary W. Adamson, Mar 28 2010: (Start)

A174739 is equivalent to Euler's pentagonal theorem in triangular form.

For example, row 9 = (0, 0, -2, 0, -5, 0, 0, 15, 22) or: p(9) = 30 = p(8)

+ p(7) - p(4) - p(2). (End)

LINKS

Table of n, a(n) for n=1..90.

FORMULA

Given triangle A145006, delete the first "1", = triangle Q. With M = A174712,

the diagonalize variant of the partition numbers, perform Q*M as infinite lower

triangular matrices.

EXAMPLE

First few rows of the triangle =

1;

1, 1;

0, 1, 2;

0, 0, 2, 3;

-1, 0, 0, 3, 5;

0, -1, 0, 0, 5, 7;

-1, 0, -2, 0, 0, 7, 11;

0, -1, 0, -3, 0, 0, 11, 15;

0, 0, -2, 0, -5, 0, 0, 15, 22;

0, 0, 0, -3, 0, -7, 0, 0, 22, 30;

0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42;

1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56;

0, 1, 0, 0, 0, 0, -11, 0, 22, 0, 0, 56, 77;

0, 0, 2, 0, 0, 0, 0, -15, 0, -30, 0, 0, 77, 101;

1, 0, 0, 3, 0, 0, 0, 0, -22, 0, -42, 0, 0, 101, 135;

0, 1, 0, 0, 5, 0, 0, 0, 0, -30, 0, -56, 0, 0, 135, 176;

0, 0, 2, 0, 0, 7, 0, 0, 0, 0, -42, 0, -77, 0, 0, 176, 231;

0, 0, 0, 3, 0, 0, 11, 0, 0, 0, 0, -56, 0, -101, 0, 0, 231, 297;

...

CROSSREFS

Cf. A000041, A145006, A174712

Sequence in context: A306418 A323886 A340829 * A280542 A340378 A274575

Adjacent sequences:  A174736 A174737 A174738 * A174740 A174741 A174742

KEYWORD

tabl,sign

AUTHOR

Gary W. Adamson, Mar 28 2010

STATUS

approved

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Last modified May 18 22:17 EDT 2021. Contains 344004 sequences. (Running on oeis4.)