A094088 -id:A094088 - cp's OEIS frontend

A255928 Expansion of exp( Sum_{n >= 1} A094088(n)*x^n/n ).

1, 1, 4, 44, 1025, 41693, 2617128, 234091692, 28251572652, 4421489003700, 870650503128708, 210629395976568828, 61405707768736724472, 21231253444779700476672, 8589776776743377081599500, 4020181599664131540547091076, 2155088041310451318611119556661

Offset: 0

Peter Bala, Mar 11 2015

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + x + 4*x^2 + 44*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A094088.
This sequence is the particular case m = 1 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255929(m = 2) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

Cf. A094088, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255929(m = 2), A255930(m = 3).

A094088 := proc (n) option remember; if n = 0 then 1 else add(binomial(2*n, 2*k)*A094088(k), k = 0 .. n-1) end if; end proc:
A255928 := proc (n) option remember; if n = 0 then 1 else add(A094088(n-k)*A255928(k), k = 0 .. n-1)/n end if; end proc:
seq(A255928(n), n = 0 .. 16);

O.g.f.: exp(x + 7*x^2/2 + 121*x^3/3 + 3907*x^4/4 + ...) = 1 + x + 4*x^2 + 44*x^3 + 1025*x^4 + ....

a(0) = 1 and a(n) = (1/n)*Sum_{k = 0..n-1} A094088(n-k)*a(k) for n >= 1.

A362586 Triangle read by rows, T(n, k) = A094088(n) * binomial(n, k).

Original entry on oeis.org

1, 1, 1, 7, 14, 7, 121, 363, 363, 121, 3907, 15628, 23442, 15628, 3907, 202741, 1013705, 2027410, 2027410, 1013705, 202741, 15430207, 92581242, 231453105, 308604140, 231453105, 92581242, 15430207, 1619195761, 11334370327, 34003110981, 56671851635, 56671851635, 34003110981, 11334370327, 1619195761

Offset: 0

Peter Luschny, Apr 26 2023

			[0]      1;
[1]      1,       1;
[2]      7,      14,       7;
[3]    121,     363,     363,     121;
[4]   3907,   15628,   23442,   15628,    3907;
[5] 202741, 1013705, 2027410, 2027410, 1013705, 202741;

Family of triangles: A055372 (m=0, Pascal), A362585 (m=1, Fubini), this sequence (m=2, Joffe), A362849 (m=3, A278073).

Cf. A094088 (column 0 and main diagonal), A362587 (row sums).

# uses[TransOrdPart from A362585]
def A362586(n) -> list[int]: return TransOrdPart(2, n)
for n in range(6): print(A362586(n))

A362587 a(n) = 2^n * A094088(n). Row sums of A362586.

Original entry on oeis.org

1, 2, 28, 968, 62512, 6487712, 987533248, 207257057408, 57359688424192, 20240182500956672, 8869195638810631168, 4725115451770644482048, 3007722163880719988764672, 2254432760608214922012434432, 1965374406868398554356767244288, 1971745067277979562424894483365888

Offset: 0

Peter Luschny, Apr 26 2023

nonn

Cf. A094088, A362586.

def A362587(n) :
    @cached_function
    def f(n) :
        if n == 0 : return 1
        if n % 2 != 0 : return 0
        return sum(f(k) * binomial(n, k) for k in range(n)[::2])
    return 2^n * f(2 * n)
print([A362587(n) for n in range(16)])

A028296 Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2.

Original entry on oeis.org

1, -1, 5, -61, 1385, -50521, 2702765, -199360981, 19391512145, -2404879675441, 370371188237525, -69348874393137901, 15514534163557086905, -4087072509293123892361, 1252259641403629865468285, -441543893249023104553682821, 177519391579539289436664789665

Offset: 0

N. J. A. Sloane

The Euler numbers A000364 with alternating signs.
The first column of the inverse to the matrix with entries C(2*i,2*j), i,j >=0. The full matrix is lower triangular with the i-th subdiagonal having entries a(i)*C(2*j,2*i) j>=i. - Nolan Wallach (nwallach(AT)ucsd.edu), Dec 26 2005
This sequence is also EulerE(2*n). - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 14 2006
Consider the sequence defined by a(0)=1; thereafter a(n) = c*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
To avoid possible confusion: these are the odd e.g.f. coefficients of Gudermannian(x) with the offset shifted by -1 (even coefficients are zero). They are identical to the even e.g.f. coefficients for 1/cosh(x) = -Gudermannian'(x) (see the Example). Since the complex root of cosh(z) with the smallest absolute value is z0 = i*Pi/2, the radius of convergence for the Taylor series of all these functions is Pi/2 = A019669. - Stanislav Sykora, Oct 07 2016

			Gudermannian(x) = x - (1/6)*x^3 + (1/24)*x^5 - (61/5040)*x^7 + (277/72576)*x^9 + ....
Gudermannian'(x) = 1/cosh(x) = (1/1!)*x^0 - (1/2!)*x^2 + (5/4!)*x^4 - (61/6!)*x^6 + (1385/8!)*x^8 + .... - _Stanislav Sykora_, Oct 07 2016

Gradshteyn and Ryzhik, Tables, 5th ed., Section 1.490, pp. 51-52.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B45.

T. D. Noe, Table of n, a(n) for n = 0..100
Beáta Bényi and Toshiki Matsusaka, Extensions of the combinatorics of poly-Bernoulli numbers, arXiv:2106.05585 [math.CO], 2021.
Filippo Callegaro and Giovanni Gaiffi, On models of the braid arrangement and their hidden symmetries, arXiv preprint arXiv:1406.1304 [math.AT], 2014.
Karl Dilcher and Christophe Vignat, Euler and the Strong Law of Small Numbers, Amer. Math. Mnthly, 123 (May 2016), 486-490.
Allan L. Edmonds and Steven Klee, The combinatorics of hyperbolized manifolds, arXiv preprint arXiv:1210.7396 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 02 2013
Guodong Liu, On congruences of Euler numbers modulo powers of two, Journal of Number Theory, Volume 128, Issue 12, December 2008, Pages 3063-3071.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924, p. 25.
Joe Santmyer, Derivative Polynomials for Trigonometric and Hyperbolic Functions, Arhimede Math. J. (2023) Vol. 10, No. 2, 152-158.
Jan W. H. Swanepoel, A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers, Integers (2025) Vol. 25, Art. No. A50. See p. 4.
Zhi-Hong Sun, On the further properties of {U_n}, arXiv:1203.5977v1 [math.NT], Mar 27 2012.

Absolute values are the Euler numbers A000364.

Cf. A019669.

A028296 := proc(n) a :=0 ; for k from 1 to 2*n+1 by 2 do a := a+(-1)^((k-1)/2)/2^k/k *add( (-1)^i *(k-2*i)^(2*n+1) *binomial(k,i), i=0..k) ; end do: a ; end proc:
seq(A028296(n),n=0..10) ; # R. J. Mathar, Apr 20 2011

Table[EulerE[2*n], {n, 0, 30}] (* Paul Abbott, Apr 14 2006 *)
Table[(CoefficientList[Series[1/Cosh[x],{x,0,40}],x]*Range[0,40]!)[[2*n+1]],{n,0,20}] (* Vaclav Kotesovec, Aug 04 2014*)
With[{nn=40},Take[CoefficientList[Series[Gudermannian[x],{x,0,nn}],x] Range[ 0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Feb 24 2018 *)
{1, Table[2*(-I)*PolyLog[-2*n, I], {n, 1, 12}]} // Flatten (* Peter Luschny, Aug 12 2021 *)
a[0] := 1; a[n_] := a[n] = -Sum[Binomial[2 n, 2 k] a[k], {k, 0, n - 1}]; Map[a, Range[0, 16]] (* Oliver Seipel, May 19 2024 *)

a(n):=sum((-1+(-1)^(k))*(-1)^((k+1)/2)/(2^(k+1)*k)*sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k),k,1,n); /* with interpolated zeros, Vladimir Kruchinin, Apr 20 2011 */

a(n) = 2*imag(polylog(-2*n, I)); \\ Michel Marcus, May 30 2018

a(n)=eulerfrac(2*n) \\ Charles R Greathouse IV, Mar 23 2022

from sympy import euler
def A028296(n): return euler(n<<1) # Chai Wah Wu, Apr 16 2023

def A028296_list(len):
    f = lambda k: x*(k+1)^2
    g = 1
    for k in range(len-2,-1,-1):
        g = (1-f(k)/(f(k)+1/g)).simplify_rational()
    return taylor(g, x, 0, len-1).list()
print(A028296_list(17))

def A028296(n):
    shapes = ([x*2 for x in p] for p in Partitions(n))
    return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
print([A028296(n) for n in (0..16)]) # Peter Luschny, Aug 10 2015

E.g.f.: sech(x) = 1/cosh(x) (even terms), or Gudermannian(x) (odd terms).
Recurrence: a(n) = -Sum_{i=0..n-1} a(i)*binomial(2*n, 2*i). - Ralf Stephan, Feb 24 2005
a(n) = Sum_{k=1,3,5,..,2n+1} ((-1)^((k-1)/2) /(2^k*k)) * Sum_{i=0..k} (-1)^i*(k-2*i)^(2n+1) * binomial(k,i). - Vladimir Kruchinin, Apr 20 2011
a(n) = 2^(4*n+1)*(zeta(-2*n,1/4) - zeta(-2*n,3/4)). - Gerry Martens, May 27 2011
From Sergei N. Gladkovskii, Dec 15 2011 - Oct 09 2013: (Start)
Continued fractions:
G.f.: A(x) = 1 - x/(S(0)+x), S(k) = euler(2*k) + x*euler(2*k+2) - x*euler(2*k)* euler(2*k+4)/S(k+1).
E.g.f.: E(x) = 1 - x/(S(0)+x); S(k) = (k+1)*euler(2*k) + x*euler(2*k+2) - x*(k+1)* euler(2*k)*euler(2*k+4)/S(k+1).
2*arctan(exp(z)) - Pi/2 = z*(1 - z^2/(G(0) + z^2)), G(k) = 2*(k+1)*(2*k+3)*euler(2*k) + z^2*euler(2*k+2) - 2*z^2*(k+1)*(2*k+3)*euler(2*k)*euler(2*k+4)/G(k+1).
G.f.: A(x) = 1/S(0) where S(k) = 1 + x*(k+1)^2/S(k+1).
G.f.: 1/Q(0) where Q(k) = 1 - x*k*(3*k-1) - x*(k+1)*(2*k+1)*(x*k^2-1)/Q(k+1).
E.g.f.:(2 - x^4/( (x^2+2)*Q(0) + 2))/(2+x^2) where Q(k) = 4*k + 4 + 1/( 1 - x^2/( 2 + x^2 + (2*k+3)*(2*k+4)/Q(k+1))).
E.g.f.: 1/cosh(x) = 8*(1-x^2)/(8 - 4*x^2 - x^4*U(0)) where U(k) = 1 + 4*(k+1)*(k+2)/(2*k+3 - x^2*(2*k+3)/(x^2 + 8*(k+1)*(k+2)*(k+3)/U(k+1)));
G.f.: 1/U(0) where U(k) = 1 - x + x*(2*k+1)*(2*k+2)/(1 + x*(2*k+1)*(2*k+2)/U(k+1));
G.f.: 1 - x/G(0) where G(k) = 1 - x + x*(2*k+2)*(2*k+3)/(1 + x*(2*k+2)*(2*k+3)/G(k+1));
G.f.: 1/Q(0), where Q(k) = 1 - sqrt(x) + sqrt(x)*(k+1)/(1-sqrt(x)*(k+1)/Q(k+1));
G.f.: (1/Q(0) + 1)/(1-sqrt(x)), where Q(k) = 1 - 1/sqrt(x) + (k+1)*(k+2)/Q(k+1);
G.f.: Q(0), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 + 1/Q(k+1)). (End)
a(n) ~ (-1)^n * (2*n)! * 2^(2*n+2) / Pi^(2*n+1). - Vaclav Kotesovec, Aug 04 2014
a(n) = 2*Im(Li_{-2n}(i)), where Li_n(x) is polylogarithm, i=sqrt(-1). - Vladimir Reshetnikov, Oct 22 2015

A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

Original entry on oeis.org

1, 1, 6, 1, 30, 90, 1, 126, 1260, 2520, 1, 510, 13230, 75600, 113400, 1, 2046, 126720, 1580040, 6237000, 7484400, 1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400, 1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000, 1, 131070, 96461910, 8203431600, 196556560200, 1882311631200, 8266953895200, 16672848192000, 12504636144000

Offset: 1

N. J. A. Sloane, Apr 22 2014

T(n,k) gives the number of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. An example is given below. Cf. A019538 and A156289. - Peter Bala, Aug 20 2014

			Triangle begins:
1,
1, 6,
1, 30, 90,
1, 126, 1260, 2520,
1, 510, 13230, 75600, 113400,
1, 2046, 126720, 1580040, 6237000, 7484400,
1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400,
1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000,
...
From _Peter Bala_, Aug 20 2014: (Start)
Row 2: [1,6]
k  Ordered set partitions of {1,2,3,4} into k blocks    Number
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1   {1,2,3,4}                                             1
2   {1,2}{3,4}, {3,4}{1,2}, {1,3}{2,4}, {2,4}{1,3},       6
    {1,4}{2,3}, {2,3}{1,4}
(End)

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
S. A. Joffe, Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 48 (1917-1920), 193-271.

Muniru A Asiru, Rows n = 1..50 of triangle, flattened

Case m=2 of the polynomials defined in A278073.
Cf. A000680 (diagonal), A094088 (row sums), A000364 (alternating row sums), A281478 (central terms), A327022 (refinement).
Diagonals give A002446, A213455, A241172, A002456.
Cf. A019538, A036969, A156289.

Flat(List([1..10],n->List([1..n],k->1/(2^(k-1))*Sum([1..k],j->(-1)^(k-j)*Binomial(2*k,k-j)*j^(2*n))))); # Muniru A Asiru, Feb 27 2019

T := proc(n,k) option remember;
if k > n then 0
elif k=0 then k^n
elif k=1 then 1
else k*(2*k-1)*T(n-1,k-1)+k^2*T(n-1,k); fi;
end: # Minor edit to make it also work in the (0,0)-offset case. Peter Luschny, Sep 03 2022
for n from 1 to 12 do lprint([seq(T(n,k), k=1..n)]); od:

T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[, 1] = 1; T[, ] = 0; Table[T[n, k], {n, 1, 9}, {k, 1, n}] (* _Jean-François Alcover, Jul 03 2019 *)

@cached_function
def A241171(n, k):
    if n == 0 and k == 0: return 1
    if k < 0 or k > n: return 0
    return (2*k^2 - k)*A241171(n - 1, k - 1) + k^2*A241171(n - 1, k)
for n in (1..6): print([A241171(n, k) for k in (1..n)]) # Peter Luschny, Sep 06 2017

T(n,k) = 0 if k <= 0 or k > n, = 1 if k=1, otherwise T(n,k) = k*(2*k-1)*T(n-1,k-1) + k^2*T(n-1,k).
Related to Euler numbers A000364 by A000364(n) = (-1)^n*Sum_{k=1..n} (-1)^k*T(n,k). For example, A000364(3) = 61 = 90 - 30 + 1.
From Peter Bala, Aug 20 2014: (Start)
T(n,k) = 1/(2^(k-1))*Sum_{j = 1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).
T(n,k) = k!*A156289(n,k) = k!*(2*k-1)!!*A036969.
E.g.f.: A(t,z) := 1/( 1 - t*(cosh(z) - 1) ) = 1 + t*z^2/2! + (t + 6*t^2)*z^4/4! + (t + 30*t^2 + 90*t^3)*z^6/6! + ... satisfies the partial differential equation d^2/dz^2(A) = D(A), where D = t^2*(2*t + 1)*d^2/dt^2 + t*(5*t + 1)*d/dt + t.
Hence the row polynomials R(n,t) satisfy the differential equation R(n+1,t) = t^2*(2*t + 1)*R''(n,t) + t*(5*t + 1)*R'(n,t) + t*R(n,t) with R(0,t) = 1, where ' indicates differentiation w.r.t. t. This is equivalent to the above recurrence equation.
Recurrence for row polynomials: R(n,t) = t*( Sum_{k = 1..n} binomial(2*n,2*k)*R(n-k,t) ) with R(0,t) := 1.
Row sums equal A094088(n) for n >= 1.
A100872(n) = (1/2)*R(n,2). (End)

A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

Original entry on oeis.org

1, 1, 2, 3, 0, -55, -480, -3157, -15232, -16623, 898560, 16316179, 194574336, 1666248025, 5418649600, -170157839685, -5164467978240, -92955464490463, -1188910801354752, -7329026447550685, 157257042777866240, 7516793832172469481, 187200588993188069376

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..456

Cf. A000111, A007289, A009189, A094088, A205571, A302397, A352250, A352251.

nmax = 22; CoefficientList[Series[1/(1 - x Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^k Binomial[n, 2 k + 1] (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(1 / (1 - x * cos(x)))) \\ Michel Marcus, Mar 10 2022

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*I^(n-k)*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1).

a(n) = Sum_{k=0..n} k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 17 2025

A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, -2, 22, -602, 30742, -2523002, 303692662, -50402079002, 11030684333782, -3077986048956602, 1066578948824962102, -449342758735568563802, 226182806795367665865622, -134065091768709178087428602, 92423044260377387363207812342, -73323347841467639992211297199002

Offset: 0

N. J. A. Sloane, Mar 28 2012

sign

The version without signs has an interpretation as a sum over marked Schröder paths. See the Josuat-Verges and Kim reference.
Consider the sequence defined by a(0)=1; thereafter a(n) = c*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Apparently a(n) = 2*(-1)^n*A002114(n). - R. J. Mathar, Mar 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..200
Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.
Zhi-Hong Sun, On the further properties of U_n, arXiv:1203.5977 [math.NT], 2012.

Cf. A002114, A028296, A094088, A210657, A210672, A210674, A210676, A249939, A255882, A241171.

A210657:=proc(n) option remember;
   if n=0 then 1
   else -2*add(binomial(2*n,2*k)*procname(n-k),k=1..floor(n)); fi;
end;
[seq(f(n),n=0..20)];
# Second program:
a := (n) -> 2*36^n*(Zeta(0,-n*2,1/6)-Zeta(0,-n*2,2/3)):
seq(a(n), n=0..15); # Peter Luschny, Mar 11 2015

nmax=20; Table[(CoefficientList[Series[1/(2*Cosh[x]-1), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)
Table[9^n EulerE[2 n, 1/3], {n, 0, 20}] (* Vladimir Reshetnikov, Jun 05 2016 *)

a(n)=polcoeff(sum(m=0, n, (2*m)!*(-x)^m/prod(k=1, m, 1-k^2*x +x*O(x^n)) ), n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Sep 17 2012

O.g.f.: Sum_{n>=0} (2*n)! * (-x)^n / Product_{k=1..n} (1 - k^2*x). - Paul D. Hanna, Sep 17 2012
E.g.f.: 1/(2*cosh(x) - 1) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. - Paul D. Hanna, Oct 30 2014
E.g.f.: cos(z/2)/cos(3z/2) = Sum_{n>=0} abs(a(n))*x^(2*n)/(2*n)!. - Olivier Gérard, Feb 12 2014
From Peter Bala, Mar 09 2015: (Start)
a(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial.
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(3*k + 1)^2).
O.g.f. as a continued fraction: 1/(1 + (3^2 - 1^2)*x/(4 + 12^2*x/(4 + (18^2 - 2^2)*x/(4 + 24^2*x/(4 + (30^2 - 2^2)*x/(4 + 36^2*x/(4 + ... ))))))) = 1 - 2*x + 22*x^2 - 602*x^3 + 30742*x^4 - .... See Josuat-Vergès and Kim, p. 23.
The expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) appears to have integer coefficients. See A255882. (End)
a(n) = 2*36^n*(zeta(-n*2,1/6)-zeta(-n*2,2/3)), where zeta(a,z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) ~ 2 * (-1)^n * (2*n)! * 3^(2*n+1/2) / Pi^(2*n+1). - Vaclav Kotesovec, Mar 14 2015
a(n) = Sum_{k=0..n} A241171(n, k)*(-2)^k. - Peter Luschny, Sep 03 2022

A243664 Number of 3-packed words of degree n.

Original entry on oeis.org

1, 1, 21, 1849, 426405, 203374081, 173959321557, 242527666641289, 514557294036701349, 1577689559404884503761, 6714435826042791310638741, 38401291553086405072860452569, 287412720357301174793668207559205, 2753382861926383584939774967275568801

Offset: 0

N. J. A. Sloane, Jun 14 2014

nonn

See Novelli-Thibon (2014) for precise definition.

Andrew Howroyd, Table of n, a(n) for n = 0..100 (terms n = 0..30 from Peter Luschny)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 16.

Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.

g := t -> (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3: series(1/(2-g(t^(1/3))),t,14): seq(((3*n)!*coeff(%,t,n)),n=0..13); # Peter Luschny, Jul 07 2015

g[t_] := (Exp[t] + 2 Exp[-t/2] Cos[Sqrt[3] t/2])/3;
a[n_] := (3n)! SeriesCoefficient[1/(2 - g[t^(1/3)]), {t, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 13 2018, after Peter Luschny *)

seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n]=sum(k=1, n, binomial(3*n, 3*k) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020

def CEN(m, len):
    f, e, r, u = [1], [1], [1], 1
    for i in (1..len-1):
        f.append(rising_factorial(u, m))
        for k in range(i-1, -1, -1):
            e[k] = (e[k]*f[i])//f[i-k]
        s = sum(e); e.append(s); r.append(s)
        u += m
    return r
A243664 = lambda len: CEN(3,len)
A243664(14) # Peter Luschny, Jul 06 2015

# Alternative
def PackedWords3(n):
    shapes = [[x**3 for x in p] for p in Partitions(n)]
    return sum([factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes])
[PackedWords3(n) for n in (0..13)] # Peter Luschny, Aug 02 2015

a(n) = (3*n)! * [t^n] 1/(2-g(t^(1/3))) with g(t) = (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3. - Peter Luschny, Jul 07 2015

a(0) = 1; a(n) = Sum_{k=1..n} binomial(3*n,3*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020

a(0)=1 prepended, more terms from Peter Luschny, Jul 06 2015

A249939 E.g.f.: 1/(5 - 4*cosh(x)).

Original entry on oeis.org

1, 4, 100, 6244, 727780, 136330084, 37455423460, 14188457293924, 7087539575975140, 4514046217675793764, 3570250394992512270820, 3433125893070920512725604, 3944372161432193963534198500, 5336301013125557989981503385444, 8396749419933421378024498580446180

Offset: 0

Paul D. Hanna, Nov 19 2014

nonn

a(n) = 4*A242858(2*n) for n>0.
a(n) = A249940(n)/3.
a(n) == 4 (mod 96) for n>0.

			E.g.f.: E(x) = 1 + 4*x^2/2! + 100*x^4/4! + 6244*x^6/6! + 727780*x^8/8! +...
where E(x) = 1/(5 - 4*cosh(x)) = -exp(x) / (2 - 5*exp(x) + 2*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 6244*x^3/3! + 727780*x^4/4! +...
where 3*A(x) = 1 + 2*exp(x)/2 + 2*exp(4*x)/2^2 + 2*exp(9*x)/2^3 + 2*exp(16*x)/2^4 + 2*exp(25*x)/2^5 + 2*exp(36*x)/2^6 + 2*exp(49*x)/2^7 +...

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A249938, A249940, A247082, A250914, A250915, A242858.

Cf. A210676, A210657, A028296, A094088, A210672, A210674.

/* E.g.f.: 1/(5 - 4*cosh(x)) */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 1/(5 - 4*cosh(X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=1, (2*n+1)\3, 2*(3*k-1)! * Stirling2(2*n+1, 3*k)))}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, (4/3)*sum(k=0, 2*n, k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: 1/3 + (2/3)*Sum_{n>=1} exp(n^2*x) / 2^n  =  Sum_{n>=0} a(n)*x^n/n!.
a(n) = (4/3) * Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) = Sum_{k=1..[(2*n+1)/3]} 2 * (3*k-1)! * Stirling2(2*n+1, 3*k) for n>0 with a(0)=3, after Vladimir Kruchinin in A242858.

A210672 a(0)=1; thereafter a(n) = 2Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, 2, 26, 842, 50906, 4946282, 704888186, 138502957322, 35887046307866, 11855682722913962, 4863821092813045946, 2425978759725443056202, 1445750991051368583278426, 1014551931766896667943384042, 828063237870027116855857421306, 777768202388460616924079724057482

Offset: 0

N. J. A. Sloane, Mar 28 2012

Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255929. - Peter Bala, Mar 13 2015
The Stirling-Bernoulli transform of Fibonacci(n+1) = 1, 1, 2, 3, 5, 8, 13, ... is 1, 0, 2, 0, 26, 0, 842, 0, 50906, 0, ... - Philippe Deléham, May 25 2015

G. C. Greubel, Table of n, a(n) for n = 0..220
Hacène Belbachir, Yahia Djemmada, On central Fubini-like numbers and polynomials, arXiv:1811.06734 [math.CO], 2018. See p. 4.

Cf. A028296, A094088, A210657, A210674, A210676, A255929, A241171.

f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];
g(2);

nmax=20; Table[(CoefficientList[Series[1/(3-2*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

a(n) ~ 2*sqrt(Pi/5) * n^(2*n+1/2) / (exp(2*n) * (log((1+sqrt(5))/2))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015
E.g.f.: 1/(3-2*cosh(x)) (even coefficients). - Vaclav Kotesovec, Mar 14 2015
a(n) = Sum_{k = 0..2*n} A163626(2*n,k)*A000045(n+1). - Philippe Deléham, May 25 2015
a(n) = Sum_{k=0..n} A241171(n, k)*2^k. - Peter Luschny, Sep 03 2022

cp's OEIS Frontend

A255928 Expansion of exp( Sum_{n >= 1} A094088(n)*x^n/n ).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

A362586 Triangle read by rows, T(n, k) = A094088(n) * binomial(n, k).

Views

Author

Keywords

Examples

Crossrefs

Programs

SageMath

A362587 a(n) = 2^n * A094088(n). Row sums of A362586.

Views

Author

Keywords

Crossrefs

Programs

SageMath

A028296 Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

PARI

Python

Sage

Sage

Formula

A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Sage

Formula

A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A210657 a(0)=1; thereafter a(n) = -2*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

A210672 a(0)=1; thereafter a(n) = 2Sum_{k=1..n} binomial(2n,2k)a(n-k).