A005799 -id:A005799 - cp's OEIS frontend

A154283 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(2n+1,i) binomial(k+2-i,2)^n, 0 <= k <= 2*(n-1).

1, 1, 4, 1, 1, 20, 48, 20, 1, 1, 72, 603, 1168, 603, 72, 1, 1, 232, 5158, 27664, 47290, 27664, 5158, 232, 1, 1, 716, 37257, 450048, 1822014, 2864328, 1822014, 450048, 37257, 716, 1, 1, 2172, 247236, 6030140, 49258935, 163809288, 242384856, 163809288, 49258935, 6030140, 247236, 2172, 1

Offset: 1

Roger L. Bagula, Jan 06 2009

From Yahia Kahloune, Jan 30 2014: (Start)
In general, let b(k,e,p) = Sum_{i=0..k} (-1)^i*binomial(e*p+1,i)*binomial(k+e-i,e)^p. Then T(n,k) = b(k,2,n).
With these coefficients we can calculate: Sum_{i=1..n} binomial(i+e-1,e)^p = Sum_{k=0..e*(p-1)} b(k,e,p)*binomial(n+e+k,e*p+k).
For example, A085438(n) = Sum_{i=1..n} binomial(1+i,2)^3 = T(3,0)*binomial(2+n,7) + T(3,1)*binomial(3+n,7) + T(3,2)*binomial(4+n,7) + T(3,3)*binomial(5+n,7) + T(3,4)*binomial(6+n,7) = (1/5040)*(90*n^7 + 630*n^6 + 1638*n^5 + 1890*n^4 + 840*n^3 - 48*n).
(End)
T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n with exactly k descents. A descent is a pair of adjacent elements with the second element less than the first. - Andrew Howroyd, May 06 2020

			Triangle begins:
  1;
  1,     4,       1;
  1,    20,      48,        20,           1;
  1,    72,     603,      1168,         603,           72,           1;
  1,   232,    5158,     27664,       47290,        27664,        5158,  232, 1;
  1,   716,   37257,    450048,     1822014,      2864328,     1822014, ...;
  1,  2172,  247236,   6030140,    49258935,    163809288,   242384856, ...;
  1,  6544, 1568215,  72338144,  1086859301,   6727188848, 19323413187, ...;
  1, 19664, 9703890, 811888600, 21147576440, 225167210712, ... ;
  ...
The T(2,1) = 4 permutations of 1122 with 1 descent are 1212, 1221, 2112, 2211. - _Andrew Howroyd_, May 15 2020

Andrew Howroyd, Table of n, a(n) for n = 1..1600 (rows 1..40)
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 12.
Helmut Prodinger, On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs , arXiv:1102.5186 [math.CO], 2011.

Columns k=0..9 are A000012, A061981, A151624, A151625, A151626, A151627, A151628, A151629, A151630, A151631.
Row sums are A000680.
Similar triangles for e=1..6: A173018 (or A008292), this sequence, A174266, A236463, A237202, A237252.
Cf. A000680, A005799, A085438, A334781.

[(&+[(-1)^j*Binomial(2*n+1,j)*Binomial(k-j+2,2)^n: j in [0..k]]): k in [0..2*n-2], n in [1..12]]; // G. C. Greubel, Jun 13 2022

A154283 := proc(n,k)
        (1-x)^(2*n+1)*add( (l*(l+1)/2)^n*x^(l-1),l=0..k+1) ;
        coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Feb 01 2013

p[x_, n_]= (1-x)^(2*n+1)*Sum[(k*(k+1)/2)^n*x^k, {k, 0, Infinity}]/x;
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n,10}]//Flatten

T(n,k)={sum(i=0, k, (-1)^i*binomial(2*n+1, i)*binomial(k+2-i, 2)^n)} \\ Andrew Howroyd, May 09 2020

def A154283(n,k): return sum((-1)^j*binomial(2*n+1, j)*binomial(k-j+2, 2)^n for j in (0..k))
flatten([[A154283(n,k) for k in (0..2*n-2)] for n in (1..12)]) # G. C. Greubel, Jun 13 2022

T(n,k) = (-1) times coefficient of x^k in (x-1)^(2*n+1) * Sum_{k>=0} (k*(k+1)/2)^n *x^(k-1).
From Yahia Kahloune, Jan 29 2014: (Start)
Sum_{i=1..n} binomial(1+i,2)^p = Sum_{k=0..2*p-2} T(p,k)*binomial(n+2+k,2*p+1).
binomial(n,2)^p = Sum_{k=0..2*p-2} T(p,k)*binomial(n+k,2*p). (End)
From Peter Bala, Dec 21 2019: (Start)
E.g.f. as a continued fraction: (1-x)/(1-x + ( 1-exp((1-x)^2*t))*x/(1-x + (1-exp(2*(1-x)^2*t))*x/(1-x + (1-exp(3*(1-x)^2*t))*x/(1-x + ... )))) =  1 + x*t + x*(x^2 + 4*x + 1)*t^2/2! + x*(x^4 + 20*x^3 + 48*x^2 + 20*x + 1)*t^3/3! + ... (use Prodinger equation 1.1).
The sequence of alternating row sums (unsigned) [1, 1, 2, 10, 104, 1816,...] appears to be A005799. (End)

Edited by N. J. A. Sloane, Jan 30 2014 following suggestions from Yahia Kahloune (among other things, the signs of all terms have been reversed).

Edited by Andrew Howroyd, May 09 2020

A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

Original entry on oeis.org

1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976, 126780706777739389745128013824

Offset: 0

N. J. A. Sloane, Don Knuth

Also central terms of the triangle in A008280. - Reinhard Zumkeller, Nov 01 2013

Conjecture: taking the sequence modulo an integer k gives an eventually purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 9 begins [1, 1, 4, 1, 7, 4, 1, 7, 4, 1, 7, ...] with an apparent period [4, 1, 7] of length 3 = phi(9)/2 beginning at a(2). - Peter Bala, May 08 2023

V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom. 1 (1992), no. 2, 197-214.
L. 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, volume 7 (1877), 157-187.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A084938, A002832. For a signed version see A099023.
Related polynomials in A098277.
A diagonal of A323834.
Cf. A005799.

a000657 n = a008280 (2 * n) n  -- Reinhard Zumkeller, Nov 01 2013

Digits := 40: rr := array(1..40,1..40): rr[1,1] := 1: for i from 1 to 39 do rr[i+1,1] := subs(x=0,diff(1+tan(x),x$i)): od: for i from 2 to 40 do for j from 2 to i do rr[i,j] := rr[i,j-1]-(-1)^i*rr[i-1,j-1]: od: od: [seq(rr[2*i-1,i],i=1..20)];
# Alternatively after Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, n): seq(a(n), n = 0..15); # Peter Luschny, Oct 27 2017

max = 20; rr[1, 1] = 1; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[1 + Tan[x], {x, i}] /. x -> 0]; For[i = 2, i <= 2*max, i++, For[j = 2, j <= i, j++, rr[i, j] = rr[i, j - 1] - (-1)^i*rr[i - 1, j - 1]]]; Table[rr[2*i - 1, i], {i, 1, max}] (* Jean-François Alcover, Jul 10 2012, after Maple *)
T[n_,0] := KroneckerDelta[n,0]; T[n_,k_] := T[n,k]=T[n,k-1]+T[n-1,n-k]; Table[T[2n,n], {n,0,16}] (* Oliver Seipel, Nov 24 2024, after Peter Luschny *)

a(n):=(-1)^(n)*sum(binomial(n,k)*euler(n+k),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */

# Algorithm of L. Seidel (1877)
def A000657_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) :
            Am += A[k]; A[k] = Am; k += e
        if e < 0 :
            R.append(A[0])
    return R
A000657_list(30)  # Peter Luschny, Apr 02 2012

Row sums of triangle, read by rows, [0, 1, 4, 9, 16, 25, 36, 49, ...] DELTA [1, 2, 6, 5, 11, 8, 16, 11, 21, 14, ...] where DELTA is Deléham's operator defined in A084938.
G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*1x/(1-1*3x/(1-2*5x/(1-2*7x/(1-3*9x/...))))). - Ralf Stephan, Sep 09 2004
G.f.: 1/G(0) where G(k) = 1 - x*(8*k^2+4*k+1) - x^2*(k+1)^2*(4*k+1)*(4*k+3)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: G(0)/(1-x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 - x*(8*k^2+4*k+1))*(1 - x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 01 2014
a(n) = (-1)^(n)*Sum_{k=0..n} C(n,k)*Euler(n+k). - Vladimir Kruchinin, Apr 06 2015
a(n) ~ 2^(4*n+5/2) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Apr 06 2015
Conjectural e.g.f. as a continued fraction: 1/(1 - (1 - exp(-2*t))/(2 - (1 - exp(-4*t))/(1 - (1 - exp(-6*t))/(2 - (1 - exp(-8*t))/(1 - ... )))) = 1 + t + 4*t^2/2! + 46*t^3/3! + .... Cf. A005799. - Peter Bala, Dec 26 2019

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001

Corrected by Sean A. Irvine, Dec 22 2010

A122193 Triangle T(n,k) of number of loopless multigraphs with n labeled edges and k labeled vertices and without isolated vertices, n >= 1; 2 <= k <= 2*n.

Original entry on oeis.org

1, 1, 6, 6, 1, 24, 114, 180, 90, 1, 78, 978, 4320, 8460, 7560, 2520, 1, 240, 6810, 63540, 271170, 604800, 730800, 453600, 113400, 1, 726, 43746, 774000, 6075900, 25424280, 61923960, 90720000, 78813000, 37422000, 7484400

Offset: 1

Vladeta Jovovic, Aug 24 2006

T(n,k) equals the number of arrangements on a line of n (nondegenerate) finite closed intervals having k distinct endpoints. See the 'IBM Ponder This' link. An example is given below. - Peter Bala, Jan 28 2018

T(n,k) equals the number of alignments of length k of n strings each of length 2. See Slowinski. Cf. A131689 (alignments of strings of length 1) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018

			Triangle begins:
  1;
  1,  6,   6;
  1, 24, 114,  180,   90;
  1, 78, 978, 4320, 8460, 7560, 2520;
  ...
From _Francisco Santos_, Nov 17 2017: (Start)
For n=3 edges and k=4 vertices there are three loopless multigraphs without isolated vertices: a path, a Y-graph, and the multigraph {12, 34, 34}. The number of labelings in each is 3!4!/a, where a is the number of automorphisms. This gives respectively 3!4!/2 = 72, 3!4!/6 = 24 and 3!4!/8 = 18, adding up to 72 + 24 + 18 = 114. (End)
From _Peter Bala_, Jan 28 2018: (Start)
T(2,3) = 6: Consider 2 (nondegenerate) finite closed intervals [a, b] and [c, d]. There are 6 arrangements of these two intervals with 3 distinct endpoints:
  ...a--b--d....  a < b = c < d
  ...a...c--b...  a < c < b = d
  ...a--d...b...  a = c < d < b
  ...a--b...d...  a = c < b < d
  ...c...a--d...  c < a < b = d
  ...c--a--b....  c < a = d < b
T(2,4) = 6: There are 6 arrangements of the two intervals with 4 distinct endpoints:
  ...a--b...c--d.....  no intersection a < b < c < d
  ...a...c...b...d...  a < c < b < d
  ...a...c--d...b....  [c,d] is a proper subset of [a,b]
  ...c...a...d...b...  c < a < d < b
  ...c...a--b...d... [a,b] is a proper subset of [c,d]
  ...c--d...a--b.....  no intersection c < d < a < b.
Sums of powers of triangular numbers:
Row 2: Sum_{i = 2..n-1} C(i,2)^2 = C(n,3) + 6*C(n,4) + 6*C(n,5);
Row 3: Sum_{i = 2..n-1} C(i,2)^3 = C(n,3) + 24*C(n,4) + 114*C(n,5) + 180*C(n,6) + 90*C(n,7). See A024166 and A085438.
exp( Sum_{n >= 1} R(n,2)*x^n/n ) = (1 + x + 19*x^2 + 1147*x^3 + 145606*x^4 + 31784062*x^5 + ... )^4
exp( Sum_{n >= 1} R(n,3)*x^n/n ) = (1 + x + 37*x^2 + 4453*x^3 + 1126375*x^4 + 489185863*x^5 + ... )^9
exp( Sum_{n >= 1} R(n,4)*x^n/n ) = (1 + x + 61*x^2 + 12221*x^3 + 5144411*x^4 + 3715840571*x^5 + ... )^16 (End)
From _Peter Bala_, Feb 04 2018: (Start)
T(3,3) = 24 alignments of length 3 of 3 strings each of length 2. Examples include
  (i) A B -    (ii) A - B
      - C D         - C D
      - E F         E F -
There are 18 alignments of type (i) with two gap characters in one of the columns (3 ways of putting 2 gap characters in a column x 2 ways to place the other letter in the row which doesn't yet have a gap character x 3 columns: there are 6 alignments of type (ii) with a single gap character in each column (3 ways to put a single gap character in the first column x 2 ways to then place a single gap character in the second column). (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (rows 1 <= n <= 100).
Peter Bala, Deformations of the Hadamard product of power series
Peter Bala, Notes on A122193
S. Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
J. Engbers and C. Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
IBM Ponder This, Jan 01 2001
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Row sums give A055203.
Cf. A078739, A005799, A019538, A131689, A154283, A299041, A087127.
For Sum_{i = 2..n} C(i,2)^k see A024166 (k = 2), A085438 - A085442 ( k = 3 thru 7).

# Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A122193 := (n,k) -> A078739(n,k)*k!/2^n:
# Displays the truncated triangle from the definition:
seq(print(seq(A122193(n,k),k=2..2*n)),n=1..6); # Peter Luschny, Mar 25 2011

t[n_, k_] := Sum[(-1)^(n - r) Binomial[n, r] StirlingS2[n + r, k], {r, 0, n}]; Table[t[n, k] k!/2^n, {n, 6}, {k, 2, 2 n}] // Flatten (* Michael De Vlieger, Nov 18 2017, after Jean-François Alcover at A078739 *)

Double e.g.f.: exp(-x)*Sum_{n>=0} exp(binomial(n,2)*y)*x^n/n!.
T(n,k) = S_{2,2}(n,k)*k!/2^n; S_{2,2} the generalized Stirling numbers A078739. - Peter Luschny, Mar 25 2011
From Peter Bala, Jan 28 2018: (Start)
T(n,k) = Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i*(i-1)/2)^n.
T(n,k) = k*(k-1)/2*( T(n-1,k) + 2*T(n-1,k-1) + T(n-1,k-2) ) for 2 < k <= 2*n with boundary conditions T(n,2) = 1 for n >= 1 and T(n,k) = 0 if (k < 2) or (k > 2*n).
n-th row polynomial R(n,x) = Sum_{i >= 2} (i*(i-1)/2)^n * x^i/(1+x)^(i+1) for n >= 1.
1/(1-x)*R(n,x/(1-x)) = Sum_{i >= 2} (i*(i-1)/2)^n*x^i for n >= 1.
R(n,x) = 1/2^n*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*F(n+k,x), where F(n,x) = Sum_{k = 0..n} k!*Stirling2(n,k)*x^k is the n-th Fubini polynomial, the n-th row polynomial of A131689.
R(n,x) = x/(1+x)*1/2^n*Sum_{k = 0..n} binomial(n,k)*F(n+k,x) for n >= 1.
The polynomials Sum_{k = 2..2*n} T(n,k)*x^(k-2)*(1-x)^(2*n-k) are the row polynomials of A154283.
A154283 * A007318 equals the row reverse of this array.
Sum_{k = 2..2*n} T(n,k)*binomial(x,k) = ( binomial(x,2) )^n. Equivalently, Sum_{k = 2..2*n} (-1)^k*T(n,k)*binomial(x+k,k) = ( binomial(x+2,2) )^n. Cf. the Worpitzky-type identity Sum_{k = 1..n} A019538(n,k)*binomial(x,k) = x^n.
Sum_{i = 2..n-1} (i*(i-1)/2)^m = Sum_{k = 2..2*m} T(m,k) * binomial(n,k+1) for m >= 1. See Examples below.
R(n,x) = x^2 o x^2 o ... o x^2 (n factors), where o is the black diamond product of power series defined in Dukes and White. Note the polynomial x o x o ... o x (n factors) is the n-th row polynomial of A019538.
x^2*R(n,-1-x) = (1+x)^2*R(n,x) for n >= 1.
R(n+1,x) = 1/2*x^2*(d/dx)^2 ((1+x)^2*R(n,x)).
The zeros of R(n,x) belong to the interval [-1, 0].
Alternating row sums equal 1, that is R(n,-1) = 1.
R(n,-2) = 4*R(n,1) = 4*A055203(n).
4^n*Sum_{k = 2..2*n} T(n,k)*(-1/2)^k appears to equal (-1)^(n+1)*A005799(n) for n >= 1.
For k a nonzero integer, the power series A(k,x) := exp( Sum_{n >= 1} 1/k^2*R(n,k)*x^n/n ) appear to have integer coefficients. See the Example section.
Sum_{k = 2..2*n} T(n,k)*binomial(x,k-2) = binomial(x,2)^n - 2*binomial(x+1,2)^n + binomial(x+2,2)^n. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane (the corresponding property also holds for the row polynomials of A019538 with a factor of x removed). (End)
From Peter Bala, Mar 08 2018: (Start)
n-th row polynomial R(n,x) = coefficient of (z_1)^2 * ... * (z_n)^2 in the expansion of the rational function 1/(1 + x - x*(1 + z_1)*...*(1 + z_n)).
The n-th row of the table is given by the matrix product P^(-1)*v_n, where P denotes Pascal's triangle A007318 and v_n is the sequence (0, 0, 1, 3^n, 6^n, 10^n, ...) regarded as an infinite column vector, where 1, 3, 6, 10, ... is the sequence of triangular numbers A000217. Cf. A087127. (End)

Definition corrected by Francisco Santos, Nov 17 2017

A210108 Left half of Poupard's triangle, A008301.

Original entry on oeis.org

1, 1, 2, 4, 8, 10, 34, 68, 94, 104, 496, 992, 1420, 1712, 1816, 11056, 22112, 32176, 40256, 45496, 47312, 349504, 699008, 1026400, 1309568, 1528384, 1666688, 1714000, 14873104, 29746208, 43920304, 56696384, 67419664, 75523808, 80571184, 82285184, 819786496

Offset: 0

Reinhard Zumkeller, Mar 17 2012

Reinhard Zumkeller, Rows n=0..120 of triangle, flattened

Cf. A002105 (left edge), A005799 (right edge); A210111.

a210108 n k = a210108_tabl !! n !! k
a210108_row n = a210108_tabl !! n
a210108_tabl = zipWith take [1..] a008301_tabf

T(n,k) = A008301(n,k), 0 <= k <= n.

A275801 Number of alternating permutations of the multiset {1,1,2,2,...,n,n}.

Original entry on oeis.org

1, 0, 1, 4, 53, 936, 25325, 933980, 45504649, 2824517520, 217690037497, 20394614883316, 2282650939846781, 300814135522967736, 46103574973075123877, 8130996533576437261772, 1635028654501420083152785, 371853339350614571322913824, 94969025880924845123887493233

Offset: 0

Max Alekseyev, Aug 09 2016

nonn

Number of permutations (p(1),...,p(2n)) of {1,1,2,2,...,n,n} satisfying p(1) < p(2) > p(3) < ... < p(2n).

a(n) <= A005799(n) <= A275829(n).

Max Alekseyev, Table of n, a(n) for n = 0..70
Wikipedia, Alternating permutation.
hkju et al., Number of updown sequences of 1,1,2,2,...,n,n, Mathoverflow, 2016.

Column k=2 of A275784.

Cf. A000111, A001250, A000459, A004075, A005799, A114938, A137729, A137730, A137737, A137749, A275829.

A130847 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,2,3,4,5,6,7,8,9,10,...] DELTA [1,1,6,6,15,15,28,28,...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 1, 1, 3, 5, 2, 15, 42, 37, 10, 105, 450, 699, 458, 104, 945, 5775, 13845, 16065, 8866, 1816, 10395, 85995, 293265, 522345, 506028, 248660, 47312

Offset: 0

Philippe Deléham, Jul 21 2007, Jul 25 2007

			Triangle begins:
1;
1, 1;
3, 5, 2;
15, 42, 37, 10;
105, 450, 699, 458, 104;
945, 5775, 13845, 16065, 8866, 1816;
10395, 85995, 293265, 522345, 506028, 248660, 47312 ;...

Sum_{k, 0<=k<=n}T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A005799(n+1), A125054(n), A126152(n), A126157(n) for x= -2, -1, 0, 1, 2, 3, 4 respectively . T(n,n)=A005799(n).

A275829 Number of weakly alternating permutations of the multiset {1,1,2,2,...,n,n}.

Original entry on oeis.org

1, 1, 2, 12, 140, 2564, 68728, 2539632, 123686800, 7677924688, 591741636128, 55438330474944, 6204888219697856, 817697605612952384, 125322509904814743424, 22102340129003429880576, 4444468680409243484516608, 1010802175212828388101900544, 258152577318424951261637001728

Offset: 0

Max Alekseyev, Aug 11 2016

nonn

Number of permutations (p(1),...,p(2n)) of {1,1,2,2,...,n,n} satisfying p(1) <= p(2) >= p(3) <= p(4) >= p(5) <= ... <= p(2n).

a(n) >= A005799(n) >= A275801(n).

Max Alekseyev, Table of n, a(n) for n = 0..70

Cf. A000111, A001250, A000459, A004075, A005799, A114938, A137729, A137730, A137737, A137749, A275801.

A272481 E.g.f. A(x,y) = cos((x - xy)/2) / cos((x + xy)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2n)y^k/n! in A(x,y), for k=0..2*n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 3, 1, 0, 0, 3, 15, 25, 15, 3, 0, 0, 17, 119, 329, 455, 329, 119, 17, 0, 0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0, 0, 2073, 22803, 110605, 311355, 563013, 683067, 563013, 311355, 110605, 22803, 2073, 0, 0, 38227, 496951, 2918825, 10241231, 23904881, 39102063, 45956625, 39102063, 23904881, 10241231, 2918825, 496951, 38227, 0, 0, 929569, 13943535, 96075665, 403075855, 1150348017, 2362479119, 3600524785, 4136759055, 3600524785, 2362479119, 1150348017, 403075855, 96075665, 13943535, 929569, 0

Offset: 0

Paul D. Hanna, May 01 2016

Row sums equal the Euler numbers, A000364.
Column 1 equals A110501, the unsigned Genocchi numbers of first kind.
Main diagonal equals A272482, where A272482(n) = A005799(n)/2^n * (2*n)!/(n!)^2.
Sum_{k=0..2*n} (-1)^k*T(n,k) = (-1)^n.
Sum_{k=0..2*n} (-2)^k*T(n,k) = 2*(-1)^n for n>0.
Sum_{k=0..2*n} 2^k*T(n,k) = (-1)^n*A210657(n).
Sum_{k=0..2*n} 3^k*T(n,k) = A000281(n).
Sum_{k=0..2*n} 4^k*T(n,k) = A272158(n).
Sum_{k=0..2*n} 2^k*3^(2*n-k)*T(n,k) = A272467(n).

			E.g.f.: A(x,y) = 1 + x^2*(y)/2! + x^4*(y + 3*y^2 + y^3)/4! +
x^6*(3*y + 15*y^2 + 25*y^3 + 15*y^4 + 3*y^5)/6! +
x^8*(17*y + 119*y^2 + 329*y^3 + 455*y^4 + 329*y^5 + 119*y^6 + 17*y^7)/8! +
x^10*(155*y + 1395*y^2 + 5325*y^3 + 11235*y^4 + 14301*y^5 + 11235*y^6 + 5325*y^7 + 1395*y^8 + 155*y^9)/10! +
x^12*(2073*y + 22803*y^2 + 110605*y^3 + 311355*y^4 + 563013*y^5 + 683067*y^6 + 563013*y^7 + 311355*y^8 + 110605*y^9 + 22803*y^10 + 2073*y^11)/12! +...
where A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2).
This triangle of coefficients of x^(2*n)*y^k/(2*n)!, k=0..2*n, begins:
[1];
[0, 1, 0];
[0, 1, 3, 1, 0];
[0, 3, 15, 25, 15, 3, 0];
[0, 17, 119, 329, 455, 329, 119, 17, 0];
[0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0];
[0, 2073, 22803, 110605, 311355, 563013, 683067, 563013, 311355, 110605, 22803, 2073, 0];
[0, 38227, 496951, 2918825, 10241231, 23904881, 39102063, 45956625, 39102063, 23904881, 10241231, 2918825, 496951, 38227, 0];
[0, 929569, 13943535, 96075665, 403075855, 1150348017, 2362479119, 3600524785, 4136759055, 3600524785, 2362479119, 1150348017, 403075855, 96075665, 13943535, 929569, 0]; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1088, of flattened triangle for rows 0..32.

Cf. A000364, A110501, A272482, A210657, A000281, A272158, A272467.

{T(n,k) = my(X=x+x*O(x^(2*n))); (2*n)!*polcoeff(polcoeff( cos((X-x*y)/2)/cos((X+x*y)/2), 2*n,x), k,y)}
for(n=0,10, for(k=0,2*n, print1(T(n,k),", "));print(""))

E.g.f.: A(x,y) = (cos(x) + cos(x*y)) / (1 + cos(x + x*y)).
E.g.f.: A(x,y) = (sin(x) + sin(x*y)) / sin(x + x*y).
E.g.f.: A(x,y) = (exp(i*x) + exp(i*x*y)) / (1 + exp(i*(x + x*y))), where i^2 = -1.
O.g.f.: 1/(1 - 1*y*x/(1 - (1+y)^2*x/(1 - (1+2*y)*(2+1*y)*x/(1 - (2+2*y)^2*x/(1 - (2+3*y)*(3+2*y)*x/(1 - (3+3*y)^2*x/(1 - (3+4*y)*(4+3*y)*x/(1 - (4+4*y)^2*x/(1 - (4+5*y)*(5+4*y)*x/(1 - (5+5*y)^2*x/(1 - ...))))))))))), a continued fraction.

A135594 a(n) = (1/2^n) * Sum_{i=0..n} (-1)^(n-i) * binomial(n,i) * A000364(i).

Original entry on oeis.org

1, 0, 1, 6, 73, 1380, 37801, 1417626, 69802993, 4369750440, 339034806001, 31935510092046, 3590398569115513, 474937566660074700, 73024143791301120601, 12914495107705743175266, 2603190607000627341985633, 593297406341867021292734160

Offset: 0

Vladeta Jovovic, Feb 25 2008

Let k be a positive integer. It appears that reducing the sequence {a(n): n >= 1} modulo k produces a periodic sequence with period a divisor of phi(k) unless k is of the form 2^j, when the period equals k. For example, modulo 7 the sequence becomes [0, 1, 6, 3, 1, 1, 0, 1, 6, 3, 1, 1, 0, 1, 6, 3, 1, 1, ...], with an apparent period of 6 = phi(7), while modulo 8 the sequence becomes [0, 1, 6, 1, 4, 1, 2, 1, 0, 1, 6, 1, 4, 1, 2, 1, 0, 1, 6, 1, 4, 1, 2, 1, ...] with an apparent period of 8. - Peter Bala, May 07 2023

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 4.2.2.(b).

Cf. A000657, A005799.

A000364 := proc(n) option remember ; (2*n)!*coeftayl(sec(x),x=0,2*n) ; end: A135594 := proc(n) add((-1)^(n-i)*binomial(n,i)*A000364(i),i=0..n)/2^n ; end: seq(A135594(n),n=0..20) ; # R. J. Mathar, Mar 14 2008
f:=sec(z): fser:=series(f,z=0,63): for n from 0 to 60 do b[n]:=factorial(n)*coeff(fser,z,n) end do: a:= proc(n) options operator, arrow: add((-1)^(n-k)*binomial(n,k)*b[2*k],k=0..n)/2^n end proc: seq(a(n),n=0..16); # Emeric Deutsch, Mar 17 2008

Table[(-1)^n*Sum[Binomial[n, k]*EulerE[2*k], {k, 0, n}]/2^n, {n, 0, 20}] (* Vaclav Kotesovec, Jun 08 2019 *)

G.f.: 1/Q(0), where Q(k)= 1 + x - x*(2*k+1)*(k+1)/(1 - x*(2*k+1)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ 2^(3*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 08 2019
Conjecture: e.g.f. as a continued fraction: 2*exp(-t)/(2 - (1-exp(-4*t))/(2 - (1-exp(-8*t))/(2 - (1-exp(-12*t))/(2 - ... )))) = 1 + t^2/2! + 6*t^3/3! + 73*t^4/4! + .... Cf. A000657 and A005799. - Peter Bala, Dec 21 2019

More terms from R. J. Mathar and Emeric Deutsch, Mar 03 2008

cp's OEIS Frontend

A088990 Triangle related to generalized Euler numbers of type 2^n (A005799).

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A154283 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(2*n+1,i) * binomial(k+2-i,2)^n, 0 <= k <= 2*(n-1).

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A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

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A122193 Triangle T(n,k) of number of loopless multigraphs with n labeled edges and k labeled vertices and without isolated vertices, n >= 1; 2 <= k <= 2*n.

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A210108 Left half of Poupard's triangle, A008301.

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A275801 Number of alternating permutations of the multiset {1,1,2,2,...,n,n}.

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A130847 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,2,3,4,5,6,7,8,9,10,...] DELTA [1,1,6,6,15,15,28,28,...] where DELTA is the operator defined in A084938 .

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A275829 Number of weakly alternating permutations of the multiset {1,1,2,2,...,n,n}.

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A154283 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(2n+1,i) binomial(k+2-i,2)^n, 0 <= k <= 2*(n-1).

A272481 E.g.f. A(x,y) = cos((x - xy)/2) / cos((x + xy)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2n)y^k/n! in A(x,y), for k=0..2*n.