cp's OEIS Frontend

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.

A014781 Seidel's triangle, read by rows.

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%I A014781 #57 Apr 27 2025 12:31:05
%S A014781 1,1,1,1,2,1,2,3,3,8,6,3,8,14,17,17,56,48,34,17,56,104,138,155,155,
%T A014781 608,552,448,310,155,608,1160,1608,1918,2073,2073,9440,8832,7672,6064,
%U A014781 4146,2073,9440,18272,25944,32008,36154,38227
%N A014781 Seidel's triangle, read by rows.
%C A014781 Named after the German mathematician Philipp Ludwig von Seidel (1821-1896). - _Amiram Eldar_, Jun 13 2021
%H A014781 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2212.07232">Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers</a>, arXiv:2212.07232 [math.CO], 2022. See p. 13.
%H A014781 Dominique Dumont and Arthur Randrianarivony, <a href="http://dx.doi.org/10.1016/0012-365X(94)90230-5">Dérangements et nombres de Genocchi</a>, Discrete Math., Vol. 132, No. 1-3 (1994), pp. 37-49.
%H A014781 Dominique Dumont and Jiang Zeng, <a href="https://doi.org/10.1023/A:1009759202242">Polynomes d'Euler et fractions continues de Stieltjes-Rogers</a>, The Ramanujan Journal, Vol. 2, No. 3 (1998), pp. 387-410; <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">alternative link</a>.
%H A014781 Richard Ehrenborg and Einar Steingrímsson, <a href="http://dx.doi.org/10.1006/eujc.1999.0370">Yet another triangle for the Genocchi numbers</a>, European J. Combin., Vol. 21, No. 5 (2000), pp. 593-600. MR1771988 (2001h:05008).
%H A014781 Evgeny Feigin, <a href="https://doi.org/10.1016/j.ejc.2012.06.004">The median Genocchi numbers, q-analogues and continued fractions</a>, European Journal of Combinatorics, Vol. 33, No. 8 (2012), pp. 1913-1918; <a href="http://arxiv.org/abs/1111.0740">arXiv preprint</a>, arXiv:1111.0740 [math.CO], 2011-2012.
%H A014781 Guo-Niu Han and Jiang Zeng, <a href="http://www.labmath.uqam.ca/~annales/volumes/23-1/PDF/063-072.pdf">On a q-sequence that generalizes the median Genocchi numbers</a>, Annal Sci. Math. Québec, Vol. 23, No. 1 (1999), pp. 63-72.
%H A014781 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%H A014781 Qiongqiong Pan and Jiang Zeng, <a href="https://arxiv.org/abs/2108.03200">Cycles of even-odd drop permutations and continued fractions of Genocchi numbers</a>, arXiv:2108.03200 [math.CO], 2021.
%H A014781 Ludwig Seidel, <a href="http://publikationen.badw.de/de/003384831">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187.
%e A014781 Triangle begins:
%e A014781      1;
%e A014781      1;
%e A014781      1,    1;
%e A014781      2,    1;
%e A014781      2,    3,    3;
%e A014781      8,    6,    3;
%e A014781      8,   14,   17,   17;
%e A014781     56,   48,   34,   17;
%e A014781     56,  104,  138,  155,  155;
%e A014781    608,  552,  448,  310,  155;
%e A014781    608, 1160, 1608, 1918, 2073, 2073;
%e A014781   9440, 8832, 7672, 6064, 4146, 2073;
%e A014781   ...
%t A014781 max = 13; T[1, 1] = 1; T[n_, k_] /; 1 <= k <= (n+1)/2 := T[n, k] = If[EvenQ[n], Sum[T[n-1, i], {i, k, max}], Sum[T[n-1, i], {i, 1, k}]]; T[_, _] = 0; Table[T[n, k], {n, 1, max}, {k, 1, (n+1)/2}] // Flatten (* _Jean-François Alcover_, Nov 18 2016 *)
%o A014781 (SageMath) # Algorithm of L. Seidel (1877)
%o A014781 # n -> Prints first n rows of the triangle
%o A014781 def A014781_triangle(n) :
%o A014781     D = []; [D.append(0) for i in (0..n)]; D[1] = 1
%o A014781     b = True
%o A014781     for i in(0..n) :
%o A014781         h = (i-1)//2 + 1
%o A014781         if b :
%o A014781             for k in range(h-1,0,-1) : D[k] += D[k+1]
%o A014781         else :
%o A014781             for k in range(1,h+1,1) :  D[k] += D[k-1]
%o A014781         b = not b
%o A014781         if i>0 : print([D[z] for z in (1..h)])
%o A014781 A014781_triangle(12) # _Peter Luschny_, Apr 01 2012
%Y A014781 Even terms of first column give A005439. Diagonal gives A001469.
%Y A014781 Cf. A005439, A001469.
%K A014781 tabf,nonn
%O A014781 1,5
%A A014781 _N. J. A. Sloane_
%E A014781 More terms from Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 18 2001