The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A154343

(Bold, blue-underlined text is an ; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

S(n,k) an additive decomposition of the Springer number (generalized Euler number), (triangle read by rows).
(history; published version)

#18 by Joerg Arndt at Mon Feb 03 01:54:41 EST 2025

STATUS

proposed

#17 by Jason Yuen at Mon Feb 03 00:28:16 EST 2025

STATUS

editing

#16 by Jason Yuen at Mon Feb 03 00:27:41 EST 2025

FORMULA

Let c(k) = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation).

S(n,k) = Sum_{v=0,..,k} ( (-1)^(v)*binomial(k,v)*2^n*c(k)*(v+3/2)^n );

S(n) = Sum_{k=0,..,n} S(n,k).

EXAMPLE

1,

3, -2,

9, -16, 4,

27, -98, 60, 0,

81, -544, 616, 0, -96,

243, -2882, 5400, 0, -3360, 960,

729, -14896, 43564, 0, -72480, 46080, -5760,

2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0,

6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120.

CROSSREFS

STATUS

approved

#15 by Bruno Berselli at Tue Sep 13 02:54:01 EDT 2016

STATUS

reviewed

#14 by Joerg Arndt at Tue Sep 13 02:48:27 EDT 2016

STATUS

proposed

#13 by G. C. Greubel at Tue Sep 13 00:41:07 EDT 2016

STATUS

editing

Discussion

Tue Sep 13

00:46

Michel Marcus: Yes, another way out, is too begin each line with a character (say a dot) and append blanks and integers behind.

01:35

Michel Marcus: To do the kind of formating you add in mind, the last line would have been to be removed. But now it looks so-so

01:48

G. C. Greubel: This is where a practice page could be useful. Triangles sometimes look good, then others get the repeat edits and end with "close enough".

#12 by G. C. Greubel at Tue Sep 13 00:39:18 EDT 2016

EXAMPLE

1,

3, -2,

9, -16, 4,

27, -98, 60, 0,

81, -544, 616, 0, -96,

243, -2882, 5400, 0, -3360, 960,

729, -14896, 43564, 0, -72480, 46080, -5760,

2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0, 6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120.

Discussion

Tue Sep 13

00:41

G. C. Greubel: It can be frustrating at times.

#11 by G. C. Greubel at Tue Sep 13 00:35:34 EDT 2016

EXAMPLE

1,

3, -2,

9, -16, 4,

27, -98, 60, 0,

81, -544, 616, 0, -96,

243, -2882, 5400, 0, -3360, 960,

729, -14896, 43564, 0, -72480, 46080, -5760,

2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0,

6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120.

STATUS

proposed

#10 by G. C. Greubel at Mon Sep 12 21:22:55 EDT 2016

STATUS

editing

Discussion

Tue Sep 13

00:26

Michel Marcus: If you enter a single space at the beginning of a line, it is removed by system when saving, So your traingle has been "broken". One way to solve this, is to have 2 blanks on the last line.

#9 by G. C. Greubel at Mon Sep 12 21:22:50 EDT 2016

LINKS

FORMULA

S(n,k) = Sum(v=0..k,(-1)^(v)*binomial(k,v)*2^n*c(k)*(v+3/2)^n);

S(n) = Sum(k=0..n, S(n,k)).

EXAMPLE

1,

3, -2,

9, -16, 4,

27, -98, 60, 0,

81, -544, 616, 0, -96,

243, -2882, 5400, 0, -3360, 960,

729, -14896, 43564, 0, -72480, 46080, -5760,

2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0,

6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120,

STATUS

approved