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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 30 21:27 EDT 2020. Contains 337440 sequences. (Running on oeis4.)