Template:Sequence of the Day for November 3

Intended for: November 3, 2012

Timetable

• First draft entered by Richard M. Green on July 30, 2011 ✓
• Draft reviewed by Daniel Forgues on November 1, 2012 ✓
• Draft to be approved by October 3, 2012

Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.

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A119258: Triangle read by rows

n   ≥   0: T  (n, 0) = T  (n, n) = 1

and for

0 < k < n: T  (n, k ) = 2 T  (n  −  1, k  −  1) + T  (n  −  1, k )

.

						1
					1		1
				1		3		1
			1		5		7		1
		1		7		17		15		1
	1		9		31		49		31		1
1		11		49		111		129		63		1

This sequence is a Pascal-like triangle (see Pascal triangle). It appears naturally in combinatorics, topology, representation theory, computer science and numerical analysis. The natural formula for the terms seems to depend heavily on the context in which they appear, but it is not hard to show by hand that all these formulae satisfy the defining recursive formula.

{1, 3, 17, 111, ...}

A119259 Central terms of the triangle in A119258.

k

th rising diagonal (read right to left): Expansion of

? + 1 (1  −  x) k  + 1  , k   ≥   0

.

{1, 3, 5, 7, ...}	A005408 The odd numbers: 2 n + 1, n   ≥   0 . Expansion of   x + 1 (1  −  x) 2 .
{1, 7, 17, 31, ...}	A056220 2 (n + 1) 2  −  1 = 2 n  (n + 2) + 1 = 2 n 2 + 4 n + 1, n   ≥   0 . Expansion of    − x 2 + 4 x + 1 (1  −  x) 3 .
{1, 15, 49, 111, ...}	A199899 2 n 3 (n + 2)  (2 n + 5) + 1 = 2 n 3 (2 n 2 + 9 n + 10) + 1, n   ≥   0 . Expansion of   x 3  −  5 x 2 + 11 x + 1 (1  −  x) 4 .
{1, 31, 129, 351, ...}	A?????? 2 n 3 (n + 2) 2  (n + 4) + 1 = 2 n 3 (n 3 + 8 n 2 + 20 n + 16) + 1, n   ≥   0 . Expansion of    − x 4 + 6 x 3  −  16 x 2 + 26 x + 1 (1  −  x) 5 .
{1, 63, 321, 1023, ...}	A?????? 2 n 15 (n + 2)  (n + 4)  (2 n 2 + 13 n + 16) + 1 = 2 n 15 (2 n 4 + 25 n 3 + 110 n 2 + 200 n + 128) + 1, n   ≥   0 . Expansion of   x 5  −  7 x 4 + 22 x 3  −  42 x 2 + 57 x + 1 (1  −  x) 6 .

k

th falling diagonal (read left to right): Expansion of

1 (1  −  x)  (1  −  2 x) k  , k   ≥   0

.

{1, 3, 7, 15, ...}	A000225 2 n +1  −  1, n   ≥   0 . (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.) Expansion of   1 (1  −  x)  (1  −  2 x)   .
{1, 5, 17, 49, ...}	A000337 n 2 n +1 + 1, n   ≥   0 . Expansion of   1 (1  −  x)  (1  −  2 x) 2   .
{1, 7, 31, 111, ...}	A055580 Bjorner–Welker sequence: 1 2 (n 2 + n + 2) 2 n +1  −  1, n   ≥   0 . Expansion of   1 (1  −  x)  (1  −  2 x) 3   .
{1, 9, 49, 209, ...}	A027608 1 6 (n 3 + 3 n 2 + 8 n) 2 n +1 + 1, n   ≥   0 . Expansion of   1 (1  −  x)  (1  −  2 x) 4   .
{1, 11, 71, 351, ...}	A?????? 1 24 (n 4 + 6 n 3 + 23 n 2 + 18 n + 24) 2 n +1  −  1, n   ≥   0 . Expansion of   1 (1  −  x)  (1  −  2 x) 5   .