A014781 Seidel's triangle, read by rows.

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, 608, 552, 448, 310, 155, 608, 1160, 1608, 1918, 2073, 2073, 9440, 8832, 7672, 6064, 4146, 2073, 9440, 18272, 25944, 32008, 36154, 38227

Offset: 1

Author

N. J. A. Sloane

Keywords

• tabf
• nonn

Comments

Named after the German mathematician Philipp Ludwig von Seidel (1821-1896). - Amiram Eldar, Jun 13 2021

Examples

			Triangle begins:
     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;
   608,  552,  448,  310,  155;
   608, 1160, 1608, 1918, 2073, 2073;
  9440, 8832, 7672, 6064, 4146, 2073;
  ...
		

Links

• Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232 [math.CO], 2022. See p. 13.
• Dominique Dumont and Arthur Randrianarivony, Dérangements et nombres de Genocchi, Discrete Math., Vol. 132, No. 1-3 (1994), pp. 37-49.
• Dominique Dumont and Jiang Zeng, Polynomes d'Euler et fractions continues de Stieltjes-Rogers, The Ramanujan Journal, Vol. 2, No. 3 (1998), pp. 387-410; alternative link.
• Richard Ehrenborg and Einar Steingrímsson, Yet another triangle for the Genocchi numbers, European J. Combin., Vol. 21, No. 5 (2000), pp. 593-600. MR1771988 (2001h:05008).
• Evgeny Feigin, The median Genocchi numbers, q-analogues and continued fractions, European Journal of Combinatorics, Vol. 33, No. 8 (2012), pp. 1913-1918; arXiv preprint, arXiv:1111.0740 [math.CO], 2011-2012.
• Guo-Niu Han and Jiang Zeng, On a q-sequence that generalizes the median Genocchi numbers, Annal Sci. Math. Québec, Vol. 23, No. 1 (1999), pp. 63-72.
• Peter Luschny, An old operation on sequences: the Seidel transform.
• Qiongqiong Pan and Jiang Zeng, Cycles of even-odd drop permutations and continued fractions of Genocchi numbers, arXiv:2108.03200 [math.CO], 2021.
• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187.

Crossrefs

Even terms of first column give A005439. Diagonal gives A001469.

Cf. A005439, A001469.

Programs

Mathematica

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 *)

SageMath

# Algorithm of L. Seidel (1877)
# n -> Prints first n rows of the triangle
def A014781_triangle(n) :
    D = []; [D.append(0) for i in (0..n)]; D[1] = 1
    b = True
    for i in(0..n) :
        h = (i-1)//2 + 1
        if b :
            for k in range(h-1,0,-1) : D[k] += D[k+1]
        else :
            for k in range(1,h+1,1) :  D[k] += D[k-1]
        b = not b
        if i>0 : print([D[z] for z in (1..h)])
A014781_triangle(12) # Peter Luschny, Apr 01 2012

Extensions

More terms from Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 18 2001