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A014781 Seidel's triangle, read by rows. 3
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 (list; graph; refs; listen; history; text; internal format)
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
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.
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.
Sequence in context: A022876 A242692 A316231 * A214500 A066016 A098068
KEYWORD
tabf,nonn
AUTHOR
EXTENSIONS
More terms from Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 18 2001
STATUS
approved

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Last modified June 22 05:36 EDT 2024. Contains 373565 sequences. (Running on oeis4.)