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, 14, 91, 364, 1001, 2002, 3003, 3432, 15, 105, 455, 1365, 3003, 5005, 6435, 6435

Offset: 1

Author

Wolfdieter Lang, May 22 2015

Keywords

• nonn
• easy
• tabf

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

Examples

			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
  ...
		

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.

Crossrefs

Cf. A007318, A258143, A258144, A014410 (Scheubel's version).

Cf. A001405 (right edge).

Programs

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

Mathematica

Table[Binomial[n, m], {n, 15}, {m, Ceiling[n/2]}] (* Paolo Xausa, Nov 14 2024 *)

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.