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

1,5

Comments

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

References

Qiongqiong Pan and Jiang Zeng, Cycles of even-odd drop permutations and continued fractions of Genocchi numbers, arXiv:2108.03200 [math.CO], 2021.

Links

Table of n, a(n) for n=1..48.

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.

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.

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

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

Cf. A005439, A001469.

Sequence in context: A022876 A242692 A316231 * A214500 A066016 A098068

Adjacent sequences: A014778 A014779 A014780 * A014782 A014783 A014784

Keyword

tabf,nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved