A257241 Irregular triangle read by rows: Stifel's version of the arithmetical triangle.

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

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

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

Adjacent sequences: A257238 A257239 A257240 * A257242 A257243 A257244

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, May 22 2015

Status

approved