This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A257241 #31 Nov 14 2024 05:30:18
%S A257241 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,
%T A257241 45,120,210,252,11,55,165,330,462,462,12,66,220,495,792,924,13,78,286,
%U A257241 715,1287,1716,1716,14,91,364,1001,2002,3003,3432,15,105,455,1365,3003,5005,6435,6435
%N A257241 Irregular triangle read by rows: Stifel's version of the arithmetical triangle.
%C A257241 The row length of this array is A008619(n-1), for n >= 1: 1, 1, 2, 2, ...
%C A257241 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.
%C A257241 The row sums give A258143. The alternating row sums give A258144.
%C A257241 T(n,A008619(n-1)) = A001405(n). - _Reinhard Zumkeller_, May 22 2015
%D A257241 H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 260.
%D A257241 Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009.
%D A257241 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.
%H A257241 Reinhard Zumkeller, <a href="/A257241/b257241.txt">Rows n = 1..150 of triangle, flattened</a>
%H A257241 MacTutor History of Mathematics archive, <a href="https://web.archive.org/web/20121021091648/http://www-history.mcs.st-andrews.ac.uk/Biographies/Apianus.html">Petrus Apianus</a>.
%H A257241 MacTutor History of Mathematics archive, <a href="https://web.archive.org/web/20080708070115/http://www-history.mcs.st-andrews.ac.uk/Biographies/Stifel.html">Michael Stifel</a>
%H A257241 Maurer, Bertram, <a href="https://web.archive.org/web/20020129003339/http://www.kk.s.bw.schule.de/mathge/stifel.htm">Michael Stifel</a>, 1999, Kolping-Kolleg Stuttgart.
%H A257241 Wikipedia, <a href="https://en.wikipedia.org/wiki/Petrus_Apianus">Petrus Apianus</a>.
%H A257241 Wikipedia, <a href="https://en.wikipedia.org/wiki/Michael_Stifel">Michael Stifel</a>.
%F A257241 T(n, m) = binomial(n, m), n >= 1, m = 1, 2, ..., ceiling(n/2).
%F A257241 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.
%e A257241 The irregular triangle T(n, m) begins:
%e A257241 n\m| 1 2 3 4 5 6 7 ...
%e A257241 ---+-------------------------------------
%e A257241 1 | 1
%e A257241 2 | 2
%e A257241 3 | 3 3
%e A257241 4 | 4 6
%e A257241 5 | 5 10 10
%e A257241 6 | 6 15 20
%e A257241 7 | 7 21 35 35
%e A257241 8 | 8 28 56 70
%e A257241 9 | 9 36 84 126 126
%e A257241 10 | 10 45 120 210 252
%e A257241 11 | 11 55 165 330 462 462
%e A257241 12 | 12 66 220 495 792 924
%e A257241 13 | 13 78 286 715 1287 1716 1716
%e A257241 ...
%t A257241 Table[Binomial[n, m], {n, 15}, {m, Ceiling[n/2]}] (* _Paolo Xausa_, Nov 14 2024 *)
%o A257241 (Haskell)
%o A257241 a257241 n k = a257241_tabf !! (n-1) !! (k-1)
%o A257241 a257241_row n = a257241_tabf !! (n-1)
%o A257241 a257241_tabf = iterate stifel [1] where
%o A257241 stifel xs@(x:_) = if odd x then xs' else xs' ++ [last xs']
%o A257241 where xs' = zipWith (+) xs (1 : xs)
%o A257241 -- _Reinhard Zumkeller_, May 22 2015
%Y A257241 Cf. A007318, A258143, A258144, A014410 (Scheubel's version).
%Y A257241 Cf. A001405 (right edge).
%K A257241 nonn,easy,tabf
%O A257241 1,2
%A A257241 _Wolfdieter Lang_, May 22 2015