login
A257241
Irregular triangle read by rows: Stifel's version of the arithmetical triangle.
4
1, 2, 3, 3, 4, 6, 5, 10, 10, 6, 15, 20, 7, 21, 35, 35, 8, 28, 56, 70, 9, 36, 84, 126, 126, 10, 45, 120, 210, 252, 11, 55, 165, 330, 462, 462, 12, 66, 220, 495, 792, 924, 13, 78, 286, 715, 1287, 1716, 1716
OFFSET
1,2
COMMENTS
The row length of this array is A008619(n-1), for n >= 1: 1, 1, 2, 2, ...
This is a truncated version of Pascal's triangle used by Michael Stifel (1487?-1567). It already appeared on the title page (frontispiece) of Peter Apianus's book of 1527 on business arithmetic: "Eyn Newe Und wolgegründte underweysung aller Kauffmanns Rechnung in dreyen Büchern". See the Kac reference, p. 394 and Table 12.1 on p. 395. It appeared in Stifel's 1553 edition of Rudolff's Coß: "Die Coß Christoffs Rudolffs. Die schönen Exemplen der Coß Durch Michael Stifel gebessert und sehr gemehrt." See the MacTutor Archive link and the Alten reference.
The row sums give A258143. The alternating row sums give A258144.
T(n,A008619(n-1)) = A001405(n). - Reinhard Zumkeller, May 22 2015
REFERENCES
H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 260.
Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009.
Reich, Karin; Michael Stifel. In: Folkerts, Menso; Eberhard Knobloch; Karin Reich: Maß, Zahl und Gewicht: Mathematik als Schlüssel zu Weltverständnis und Weltbeherrschung. Wolfenbüttel 1989, S. 73 - 95 und 373.
LINKS
MacTutor History of Mathematics archive, Petrus Apianus.
MacTutor History of Mathematics archive, Michael Stifel
Maurer, Bertram, Michael Stifel, 1999, Kolping-Kolleg Stuttgart.
Wikipedia, Petrus Apianus.
Wikipedia, Michael Stifel.
FORMULA
T(n, m) = binomial(n, m), n >= 1, m = 1, 2, ..., ceiling(n/2).
O.g.f. row m = 1, 2, ..., 4 (with leading zeros): x/(1-x)^2, x^3*(3-3*x+x^2)/(1-x)^3, x^5*(10-20*x+15*x^2-4*x^3)/(1-x)^4, x^7*(35-105*x+126*x^2-70*x^3+15*x^4)/(1-x)^5.
EXAMPLE
The irregular triangle T(n, m) begins:
n\m| 1 2 3 4 5 6 7 ...
---+-------------------------------------
1 | 1
2 | 2
3 | 3 3
4 | 4 6
5 | 5 10 10
6 | 6 15 20
7 | 7 21 35 35
8 | 8 28 56 70
9 | 9 36 84 126 126
10 | 10 45 120 210 252
11 | 11 55 165 330 462 462
12 | 12 66 220 495 792 924
13 | 13 78 286 715 1287 1716 1716
...
PROG
(Haskell)
a257241 n k = a257241_tabf !! (n-1) !! (k-1)
a257241_row n = a257241_tabf !! (n-1)
a257241_tabf = iterate stifel [1] where
stifel xs@(x:_) = if odd x then xs' else xs' ++ [last xs']
where xs' = zipWith (+) xs (1 : xs)
-- Reinhard Zumkeller, May 22 2015
CROSSREFS
Cf. A007318, A258143, A258144, A014410 (Scheubel's version).
Cf. A001405 (right edge).
Sequence in context: A304705 A131187 A099072 * A239964 A290585 A106464
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, May 22 2015
STATUS
approved