A005439 -id:A005439 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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;
  ...

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.

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

Cf. A005439, A001469.

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 *)

# 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

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

A098435 Triangle of Salie numbers T(n,k) for negative n,k, n < k.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -8, 13, -6, 1, 56, -92, 45, -10, 1, -608, 1000, -493, 115, -15, 1, 9440, -15528, 7662, -1799, 245, -21, 1, -198272, 326144, -160944, 37817, -5180, 462, -28, 1, 5410688, -8900224, 4392080, -1032088, 141465, -12684, 798, -36, 1

Offset: 1

Ralf Stephan, Sep 08 2004

Inverse matrix of A054142. - Paul Barry, Jan 21 2005

Essentially the same as the triangle giving by [0,-1,-1,-4,-4,-9,-9,-16,-16,-25,...] DELTA[1,0,1,0,1,0,1,0,1,0,...] = 1; 0,1; 0,-1,1; 0,2,-3,1; 0,-8,13,-6,1; 0,56,-92,45,-10,1; ... where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 30 2006

			   1;
  -1,   1;
   2,  -3,   1;
  -8,  13,  -6,   1;
  56, -92,  45, -10,  1;

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

T(-1, k) = (-1)^k*A005439(k-1). Row sums are zero.

rows = 9; A054142 = Table[ PadRight[ Table[ Binomial[2*n-k, k], {k, 0, n}], rows], {n, 0, rows-1}]; inv = Inverse[A054142]; Table[ Take[inv[[n]], n], {n, 1, rows}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Paul Barry *)

See A065547 for formulas.

A297703 The Genocchi triangle read by rows, T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 8, 14, 17, 17, 56, 104, 138, 155, 155, 608, 1160, 1608, 1918, 2073, 2073, 9440, 18272, 25944, 32008, 36154, 38227, 38227, 198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569, 5410688, 10623104, 15448416, 19716064, 23281432, 26031912

Offset: 0

Peter Luschny, Jan 03 2018

			The triangle starts:
0: [     1]
1: [     1,      1]
2: [     2,      3,      3]
3: [     8,     14,     17,     17]
4: [    56,    104,    138,    155,    155]
5: [   608,   1160,   1608,   1918,   2073,   2073]
6: [  9440,  18272,  25944,  32008,  36154,  38227,  38227]
7: [198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569]

Row sums are A005439 with offset 0.
T(n,0) = A005439 with A005439(0) = 1.
T(n,n) = A110501 with offset 0.
Cf. A001469, A014781, A099959, A226158.

function A297703Triangle(len::Int)
    A = fill(BigInt(0), len+2); A[2] = 1
    for n in 2:len+1
        for k in n:-1:2 A[k] += A[k+1] end
        for k in 2: 1:n A[k] += A[k-1] end
        println(A[2:n])
    end
end
println(A297703Triangle(9))

from functools import cache
@cache
def T(n):  # returns row n
    if n == 0: return [1]
    row = [0] + T(n - 1) + [0]
    for k in range(n, 0, -1): row[k] += row[k + 1]
    for k in range(2, n + 2): row[k] += row[k - 1]
    return row[1:]
for n in range(9): print(T(n))  # Peter Luschny, Jun 03 2022

A058942 Triangle of coefficients of Gandhi polynomials.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 8, 22, 20, 6, 56, 184, 224, 120, 24, 608, 2248, 3272, 2352, 840, 120, 9440, 38080, 62768, 54336, 26208, 6720, 720, 198272, 856480, 1550528, 1531344, 896064, 312480, 60480, 5040, 5410688, 24719488, 48207488, 52633344, 35371776

Offset: 1

David W. Wilson, Jan 12 2001

(1+x)^2 divides these polynomials for n > 2. - T. D. Noe, Jan 01 2008

			Triangle starts:
[1]
[1,      1]
[2,      4,      2]
[8,      22,     20,      6]
[56,     184,    224,     120,     24]
[608,    2248,   3272,    2352,    840,    120]
[9440,   38080,  62768,   54336,   26208,  6720,   720]
[198272, 856480, 1550528, 1531344, 896064, 312480, 60480, 5040]

T. D. Noe, Rows n = 1..50 of triangle, flattened

First column is A005439, as are row sums. See also A036970.

Cf. A084938.

c[1][x_] = 1; c[n_][x_] :=  c[n][x] = (x+1)*((x+1)*c[n-1][x+1] - x*c[n-1][x]); Table[ CoefficientList[ c[n][x], x], {n, 9}] // Flatten (* Jean-François Alcover, Oct 09 2012 *)

# uses[delehamdelta from A084938]
def A058942_triangle(n) :
    A = [((i+1)//2)^2 for i in (1..n)]
    B = [((i+1)//2) for i in (1..n)]
    return delehamdelta(A, B)
A058942_triangle(10) # Peter Luschny, Nov 09 2019

C_1(x) = 1; C_n(x) = (x+1)*((x+1)*C_n-1(x+1) - x*C_n-1(x)).

Triangle T(n, k), read by rows; given by [1, 1, 4, 4, 9, 9, 16, 16, 25, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 24 2005

A168488 Hankel transform of Genocchi medians.

Original entry on oeis.org

1, 1, 16, 20736, 6879707136, 1426576071720960000, 383375999244747512217600000000, 247370021455402476126653493805056000000000000

Offset: 0

Paul Barry, Nov 27 2009

Hankel transform of A005439 (when this starts 1,1,2,8,...).

Cf. A000178.

Table[Product[k!^4, {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 23 2023 *)

a(n)=Product{k=0..n, ((k+1)^4)^(n-k)}.
From Vaclav Kotesovec, Nov 23 2023: (Start)
a(n) = Product_{k=0..n} k!^4.
a(n) ~ (2*Pi)^(2*n + 2) * n^(2*n^2 + 4*n + 5/3) / (A^4 * exp(3*n^2 + 4*n - 1/3)), where A is the Glaisher-Kinkelin constant A074962. (End)

A221971 G.f.: A(x,y) = Sum_{n>=0} n! * x^ny^n Product_{k=1..n} (1 + kx) / (1 + kx*y + k^2x^2y).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 4, 1, 0, 0, 3, 11, 1, 0, 0, 0, 27, 26, 1, 0, 0, 0, 17, 148, 57, 1, 0, 0, 0, 0, 278, 646, 120, 1, 0, 0, 0, 0, 155, 2590, 2481, 247, 1, 0, 0, 0, 0, 0, 4073, 18304, 8805, 502, 1, 0, 0, 0, 0, 0, 2073, 58427, 109699, 29682, 1013, 1, 0, 0, 0

Offset: 0

Paul D. Hanna, Feb 01 2013

			Triangle begins:
1;
0, 1;
0, 1, 1;
0, 0, 4, 1;
0, 0, 3, 11, 1;
0, 0, 0, 27, 26, 1;
0, 0, 0, 17, 148, 57, 1;
0, 0, 0, 0, 278, 646, 120, 1;
0, 0, 0, 0, 155, 2590, 2481, 247, 1;
0, 0, 0, 0, 0, 4073, 18304, 8805, 502, 1;
0, 0, 0, 0, 0, 2073, 58427, 109699, 29682, 1013, 1;
0, 0, 0, 0, 0, 0, 80712, 614819, 590254, 96648, 2036, 1;
0, 0, 0, 0, 0, 0, 38227, 1665829, 5340996, 2948040, 307255, 4083, 1; ...

Paul D. Hanna, Rows n = 0..32, flattened.

Cf. A208237 (row sums), A110501 (central terms), A005439 (column sums), A136126.

{T(n,k)=polcoeff(polcoeff(sum(m=0, n,m!*x^m*y^m*prod(k=1, m, (1+k*x)/(1+k*x*y+k^2*x^2*y +x*O(x^n)))),n,x),k,y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Row sums equal A208237.
Central terms equal A110501, the Genocchi numbers of first kind (unsigned).
Columns sums equal A005439, the Genocchi numbers of second kind.

A240485 a(n) = -Zeta(1-n)n(2^(n+1) - 4) - Zeta(-n)(n+1)(2^(n+2) - 2), for n = 0 the limit is understood.

Original entry on oeis.org

1, 3, 2, -1, -2, 3, 6, -17, -34, 155, 310, -2073, -4146, 38227, 76454, -929569, -1859138, 28820619, 57641238, -1109652905, -2219305810, 51943281731, 103886563462, -2905151042481, -5810302084962, 191329672483963, 382659344967926, -14655626154768697

Offset: 0

Paul Curtz, Apr 06 2014

sign

Let G(m, n) denote the difference table of a(n):
1,    3,   2,  -1,  -2,   3,   6, -17, -34,...
2,   -1,  -3,  -1,   5,   3, -23, -17,...
-3,  -2,   2,   6,  -2, -26,   6,...
1,    4,   4,  -8, -24,  32,...
3,    0, -12, -16,  56,...
-3, -12,  -4,  72,...
-9,   8,  76,...
17,  68,...
51,...
a(n) = G(0, n).
The main diagonal G(n, n) = 1, -1, 2, -8, 56, -608,... is essentially a signed version of A005439.
The first upper diagonal is the main diagonal multiplied by 3. G(n, n+1) = 3*G(n, n).
G(m, n) = G(m, n-1) + G(m+1,n-1).
Inverse binomial transform: after 1, 2, -3, A110501(n+1) is interleaved with 3*A110501(n+1), signed two by two. I. e. b(n) = 1, 2, -3, 1, 3, -3, -9, 17, 51,... . a(n+2) + b(n+2) = -1, 0, 1, 0, -3, 0, 17,... = A226158(n+2).
This is particular to the Genocchi numbers. If the first upper diagonal is proportional to the main diagonal (1, -1, 2, -8,...), the sequence and the inverse binomial transform are simply connected to the Genocchi numbers.

Cf. A001469, A005439, A110501, A226158, A230324.

A240485 := proc(n) if n = 0 then 1 elif n = 1 then 3 else
m := 2*iquo(n-1, 2) + 2; -2^irem(n-1, 2)*m*euler(m-1, 0) fi end:
seq(A240485(n), n=0..27); # Peter Luschny, Apr 09 2014

a[n_] := Which[n == 0, 1, n == 1, 3, True, m = 2*Quotient[n-1, 2]+2; -2^Mod[n-1, 2]*m*EulerE[m-1, 0]]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Apr 09 2014, after Peter Luschny *)

def A240485(n):
    if n < 3: return [1,3,2][n]
    m = 2*((n+1)//2)
    b = 2*(1-2^m)*bernoulli(m)
    if is_even(n): b = 2*b
    return (-1)^ceil((n^2+1)/2)*b
[A240485(n) for n in (0..24)]  # Peter Luschny, Apr 08 2014

a(2*n+1) = a(2*n+2)/2 for n > 0.
-a(2*n+2)/2 = A226158(2*n+2) = A001469(n+1) = (2*n+2)*E(2*n+1, 0) where E(n, x) are the Euler polynomials.
a(n) = -2*A226158(n) - A226158(n+1).
E.g.f.: (2*exp(x)*(3*x+exp(x)*(2*x+1)+1))/(exp(x)+1)^2. - Peter Luschny, Apr 10 2014

A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 0, 3, 5, 1, 0, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 0, 1, 10, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 15, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 21, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 28, 10, 1

Offset: 0

Paul Curtz, Oct 24 2016

In other words, for n > 0 the column k lists 2*k+1 zeros together with the partial sums of the positive terms of column k-1. - Omar E. Pol, Oct 25 2016
Comments from the author:
1) ZSPEC =
  0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, 0, ...
  1, 2, 0, 0, 0, 0, 0, 0, ...
  1, 3, 1, 0, 0, 0, 0, 0, ...
  1, 4, 3, 0, 0, 0, 0, 0, ...
  1, 5, 6, 1, 0, 0, 0, 0, ...
etc.
The columns are the autosequences of the first kind of the title (column 1: 0, 0, followed by A001477(n); column 2: 0, 0, 0, 0, followed by A000217(n), etc) .
The positive terms are the Pascal triangle written by diagonals (A011973).
First column: A060576(n+1). Or A057427(n), n>-1, thanks to Omar E. Pol.
Row sums: A000045(n), autosequence of the first kind.
Alternated row sums and subtractions: 0, 1, 1, 0, -1, -1, 0 = A128834(n), autosequence of the first kind.
Antidiagonal sums: 0, 1, 1, 1, 2, 3, 4, 6, ... = A078012(n+2).
Application.
Numbers in triangle leading to the Genocchi numbers -A226158(n).
We multiply the columns of ZSPEC by d(n) = 1, -1, 2, -8, 56, -608, ... from A005439.
Hence, with only the first 0,
  0,
  1,
  1,
  1, -1,
  1, -2,
  1, -3,  2,
  1, -4,  6,
  1, -5, 12,   -8,
  1, -6, 20,  -32,
  1, -7, 30,  -80,  56,
  1, -8, 42, -160, 280,
  etc.
The row sums is -A226158(n).
2) Now consider the case of the autosequences of the second kind.
First step.
            2, 1, 1,  1,  1,  1, ...        = A054977(n)
      0, 0, 2, 3, 4,  5,  6,  7, ...        = A199969(n) with offset 0
0, 0, 0, 0, 2, 5, 9, 14, 20, 27, ...        see A000096
etc.
The positive terms are ASPEC in A191302. By triangle, they are either A029653(n) with A029653(0) = 2 instead of 1 or A029635(n).
Second step. YSPEC =
  2, 0,  0, 0, 0, 0, ...
  1, 0,  0, 0, 0, 0, ...
  1, 2,  0, 0, 0, 0, ...
  1, 3,  0, 0, 0, 0, ...
  1, 4,  2, 0, 0, 0, ...
  1, 5,  5, 0, 0, 0, ...
  1, 6,  9, 2, 0, 0, ...
  1, 7, 14, 7, 0, 0, ...
  etc.
Diagonals by triangle: A029635(n).
This is the companion to ZSPEC.
Row sums: A000032(n), autosequence of the second kind.
Alternated row sums and subtractions: period 6 repeat 2, 1, -1, -2, -1, 1 = A087204(n), autosequence of the second kind.
Application.
Numbers in triangle leading to A230324(n), a companion to -A226158(n).
We multiply the columns of YSPEC by d(n) 1, -1, 2, -8, 56, ... (see above).
Hence, without zeros:
  2,
  1,
  1, -2,
  1, -3,
  1, -4,  4,
  1, -5, 10,
  1, -6, 18,  -16,
  1, -7, 28,  -56,
  1, -8, 40, -128, 112,
  1, -9, 54, -240, 504,
  etc.
The row sum is A230324(n).

G. C. Greubel, Table of n, a(n) for the first 25 antidiagonals, flattened
OEIS Wiki, Autosequence

Cf. A011973 (without 0's), A007318 (Pascal's triangle).
Cf. A000045 (row sums), A078012 (antidiagonal sums).
Columns: A060576 or A057427 (k=0), A001477 (k=1), A000217 (k=2).
Cf. A000004, A000012, A000027, A000032, A005439, A029635, A029653, A054977, A087204, A128834, A191302, A199969, A226158, A230324.

kMax = 13; col[0] = Join[{0}, Array[1&, kMax]]; col[k_] := col[k] = Join[{0, 0}, col[k-1][[1 ;; -3]] // Accumulate]; T[n_, k_] := col[k][[n+1]]; Table[T[n-k, k], {n, 0, kMax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 15 2016 *)

Better definition from Omar E. Pol, Oct 25 2016

A097418 Triangle of coefficients of a certain sequence of polynomials f_n(x) arising in connection with deformations of coordinate rings of type D Kleinian singularities.

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 8, 8, 0, 5, 20, 56, 56, 0, 6, 40, 216, 608, 608, 0, 7, 70, 616, 3352, 9440, 9440, 0, 8, 112, 1456, 12928, 70400, 198272, 198272, 0, 9, 168, 3024, 39696, 352768, 1921152, 5410688, 5410688, 0, 10, 240, 5712, 103872, 1364800, 12129664, 66057856, 186043904, 186043904, 0

Offset: 1

Paul Boddington, Aug 20 2004

f_n(x) has the property that whenever (a,b) is a pair of complex numbers satisfying 2ab = a^2 + 2a + b^2 + 2b, we have f_n(a) + f_n(b) = 2(a^n - b^n)/(a-b) (interpreted as 2na^(n-1) if a=b). Using the pairs (0,0), (0,-2), (-2,-6), (-6,-12), (-12,-20), ...(see A002378), this enables us to successively deduce the values of f_n(0), f_n(-2),... (and this of course determines f_n(x)). There may be no simpler characterization.

			The array begins
  1
  2 0
  3 2 0
  4 8 8 0
corresponding to the polynomials f_1(x) = 1, f_2(x) = 2x, f_3(x) = 3x^2 + 2x, f_4(x) = 4x^3 + 8x^2 + 8x.

Paul Boddington, No-cycle algebras and representation theory, Ph.D. thesis, University of Warwick, 2004.

Cf. A002378, A005439 (right diagonal).

f(n, a, b) = if (a==b, 2*n*a^(n-1), 2*(a^n - b^n)/(a-b));
row(n) = if (n==1, return([1])); my(v = vector(n-1, k, f(n, -(k-1)*k, -k*(k+1)))); my(m = matrix(n-1, n-1, i, j, (-j*(j-1))^i + (-(j+1)*j)^i)); concat(Vecrev(v/m), 0); \\ Michel Marcus, Mar 19 2023

Typo corrected and extended by Michel Marcus, Mar 19 2023

A193550 O.g.f.: 1/(1 - x/(1 - x/(1 - 9x/(1 - 9x/(1 - 25x/(1 - 25x/(1 - 49x/(1 - 49x/(1-...))))))))), a continued fraction involving the odd squares.

Original entry on oeis.org

1, 1, 2, 13, 206, 5794, 252068, 15663997, 1316280854, 143694972886, 19764465724412, 3343418236081618, 682133942067492236, 165161123584687819684, 46817735849074712020808, 15358634840651231221695517, 5772973821383087169122348774

Offset: 0

Paul D. Hanna, Jul 30 2011

nonn

Conjecture: given o.g.f. A(x), then sqrt( (A(x) - 1)/x ) is an integer series.

			O.g.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 206*x^4 + 5794*x^5 + 252068*x^6 +...
Let A(x) = 1 + x*B(x)^2, then B(x) appears to be an integer series:
B(x) = 1 + x + 6*x^2 + 97*x^3 + 2782*x^4 + 122670*x^5 + 7687932*x^6 + 649446621*x^7 + 71136143478*x^8 + 9806113041658*x^9 +...
the coefficients of B(x) are integer for at least the initial 500 terms.

Vaclav Kotesovec, Table of n, a(n) for n = 0..240

Cf. A005439.

nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[(2*Range[nmax + 1] - 2*Floor[Range[nmax + 1]/2] - 1)^2*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

/* Continued fraction: */
{a(n)=local(A=1+x, CF); CF=1+x; for(k=0, n, CF=1/(1-(2*((n-k)\2)+1)^2*x*CF+x*O(x^n))); A=CF; polcoeff(A, n)}

G.f.: 1/Q(0), where Q(k) = 1 -(2*k+1)^2*x/(1 -(2*k+1)^2*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 17 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)^2/( x*(2*k+1)^2 - 1/(1 - x*(2*k+1)^2/( x*(2*k+1)^2 - 1/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013
a(n) ~ 2^(4*n+2) * n^(2*n-1/2) / (Pi^(2*n+1/2) * exp(2*n)). - Vaclav Kotesovec, Aug 25 2017

cp's OEIS Frontend

A014781 Seidel's triangle, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

SageMath

Extensions

A098435 Triangle of Salie numbers T(n,k) for negative n,k, n < k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A297703 The Genocchi triangle read by rows, T(n,k) for n>=0 and 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Julia

Python

A058942 Triangle of coefficients of Gandhi polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A168488 Hankel transform of Genocchi medians.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A221971 G.f.: A(x,y) = Sum_{n>=0} n! * x^n*y^n * Product_{k=1..n} (1 + k*x) / (1 + k*x*y + k^2*x^2*y).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A240485 a(n) = -Zeta(1-n)*n*(2^(n+1) - 4) - Zeta(-n)*(n+1)*(2^(n+2) - 2), for n = 0 the limit is understood.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.

Views

Author

Keywords

Comments

A221971 G.f.: A(x,y) = Sum_{n>=0} n! * x^ny^n Product_{k=1..n} (1 + kx) / (1 + kx*y + k^2x^2y).

A240485 a(n) = -Zeta(1-n)n(2^(n+1) - 4) - Zeta(-n)(n+1)(2^(n+2) - 2), for n = 0 the limit is understood.

A193550 O.g.f.: 1/(1 - x/(1 - x/(1 - 9x/(1 - 9x/(1 - 25x/(1 - 25x/(1 - 49x/(1 - 49x/(1-...))))))))), a continued fraction involving the odd squares.