A111884 -id:A111884 - cp's OEIS frontend

A111596 The matrix inverse of the unsigned Lah numbers A271703.

1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1, 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Also the associated Sheffer triangle to Sheffer triangle A111595.
Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-1,x), which equals (-1)^n * Lag(n,x,-1) below. Lag(n,Lag(.,x,-1),-1) = x^n evaluated umbrally, i.e., with (Lag(.,x,-1))^k = Lag(k,x,-1). - Tom Copeland, Apr 26 2014
Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297.
The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.
The row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m, together with the row polynomials s(n,x) of A111595 satisfy the exponential (or binomial) convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), n>=0.
Exponential Riordan array [1,x/(1+x)]. Inverse of the exponential Riordan array [1,x/(1-x)], which is the unsigned version of A111596. - Paul Barry, Apr 12 2007
For the unsigned subtriangle without column number m=0 and row number n=0, see A105278.
Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.
The unsigned row polynomials are Lag(n,-x,-1), the associated Laguerre polynomials of order -1 with negated argument. See Gradshteyn and Ryzhik, Abramowitz and Stegun and Rota (Finite Operator Calculus) for extensive formulas. - Tom Copeland, Nov 17 2007, Sep 09 2008
An infinitesimal matrix generator for unsigned A111596 is given by A132792. - Tom Copeland, Nov 22 2007
From the formalism of A132792 and A133314 for n > k, unsigned A111596(n,k) = a(k) * a(k+1)...a(n-1) / (n-k)! = a generalized factorial, where a(n) = A002378(n) = n-th term of first subdiagonal of unsigned A111596. Hence Deutsch's remark in A002378 provides an interpretation of A111596(n,k) in terms of combinations of certain circular binary words. - Tom Copeland, Nov 22 2007
Given T(n,k)= A111596(n,k) and matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 27 2008
Operationally, the unsigned row polynomials may be expressed as p_n(:xD:) = x*:Dx:^n*x^{-1}=x*D^nx^n*x^{-1}= n!*binomial(xD+n-1,n) = (-1)^n n! binomial(-xD,n) = n!L(n,-1,-:xD:), where, by definition, :AB:^n = A^nB^n for any two operators A and B, D = d/dx, and L(n,-1,x) is the Laguerre polynomial of order -1. A similarity transformation of the operators :Dx:^n generates the higher order Laguerre polynomials, which can also be expressed in terms of rising or falling factorials or Kummer's confluent hypergeometric functions (cf. the Mathoverflow post). -  Tom Copeland, Sep 21 2019

			Binomial convolution of row polynomials: p(3,x) = 6*x-6*x^2+x^3; p(2,x) = -2*x+x^2, p(1,x) = x, p(0,x) = 1,
together with those from A111595: s(3,x) = 9*x-6*x^2+x^3; s(2,x) = 1-2*x+x^2, s(1,x) = x, s(0,x) = 1; therefore
9*(x+y)-6*(x+y)^2+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = (6*y-6*y^2+y^3) + 3*x*(-2*y+y^2) + 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.
From _Wolfdieter Lang_, Apr 28 2014: (Start)
The triangle a(n,m) begins:
n\m  0     1       2       3      4     5   6  7
0:   1
1:   0     1
2:   0    -2       1
3:   0     6      -6       1
4:   0   -24      36     -12      1
5:   0   120    -240     120    -20     1
6:   0  -720    1800   -1200    300   -30   1
7:   0  5040  -15120   12600  -4200   630 -42  1
...
For more rows see the link.
(End)

G. C. Greubel, Rows n=0..100 of triangle, flattened
Wolfdieter Lang, The first 11 rows of the triangle.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 20.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers; Generators, Inversion, and Matrix, Binomial, and Integral Transforms; Lagrange a la Lah
A. Hennessy and P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Mathoverflow, Pochhammer symbol of a differential, and hypergeometric polynomials, a question posed by Emilio Pisanty and answered by Tom Copeland, 2012.
J. Taylor, Counting words with Laguerre polynomials, DMTCS Proc., Vol. AS, 2013, p. 1131-1142. [_Tom Copeland_, Jan 08 2016] [Broken link]
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96. [_Tom Copeland_, Dec 20 2018]
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Row sums: A111884. Unsigned row sums: A000262.
A002868 gives maximal element (in magnitude) in each row.
Cf. A130561 for a natural refinement.
Cf. A264428, A264429, A271703 (unsigned).
Cf. A008297, A089231, A105278 (variants).
Cf. A002378, A008277, A132792, A133314, A132792, A001263.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::odd, -(n+1)!, (n+1)!), 9); # Peter Luschny, Jan 27 2016

a[0, 0] = 1; a[n_, m_] := ((-1)^(n-m))*(n!/m!)*Binomial[n-1, m-1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
T[ n_, k_] := (-1)^n n! Coefficient[ LaguerreL[ n, -1, x], x, k]; (* Michael Somos, Dec 15 2014 *)
rows = 9;
t = Table[(-1)^(n+1) n!, {n, 1, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}]  // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

{T(n, k) = if( n<1 || k<1, n==0 && k==0, (-1)^n * n! * polcoeff( sum(k=1, n, binomial( n-1, k-1) * (-x)^k / k!), k))}; /* Michael Somos, Dec 15 2014 */

lah_number = lambda n, k: factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)
A111596_row = lambda n: [(-1)^(n-k)*lah_number(n, k) for k in (0..n)]
for n in range(10): print(A111596_row(n)) # Peter Luschny, Oct 05 2014

# uses[inverse_bell_transform from A264429]
def A111596_matrix(dim):
    fact = [factorial(n) for n in (1..dim)]
    return inverse_bell_transform(dim, fact)
A111596_matrix(10) # Peter Luschny, Dec 20 2015

E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.
E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).
a(n, m) = ((-1)^(n-m))*|A008297(n, m)| = ((-1)^(n-m))*(n!/m!)*binomial(n-1, m-1), n>=m>=1; a(0, 0)=1; else 0.
a(n, m) = -(n-1+m)*a(n-1, m) + a(n-1, m-1), n>=m>=0, a(n, -1):=0, a(0, 0)=1; a(n, m)=0 if n|a(n,m)| = Sum_{k=m..n} |S1(n,k)|*S2(k,m), n>=0. S2(n,m):=A048993. S1(n,m):=A048994. - Wolfdieter Lang, May 04 2007
From Tom Copeland, Nov 21 2011: (Start)
For this Lah triangle, the n-th row polynomial is given umbrally by
  (-1)^n n! binomial(-Bell.(-x),n), where Bell_n(-x)= exp(x)(xd/dx)^n exp(-x), the n-th Bell / Touchard / exponential polynomial with neg. arg., (cf. A008277). E.g., 2! binomial(-Bell.(-x),2) = -Bell.(-x)*(-Bell.(-x)-1) = Bell_2(-x)+Bell_1(-x) = -2x+x^2.
A Dobinski relation is (-1)^n n! binomial(-Bell.(-x),n)= (-1)^n n! e^x Sum_{j>=0} (-1)^j binomial(-j,n)x^j/j!= n! e^x Sum_{j>=0} (-1)^j binomial(j-1+n,n)x^j/j!. See the Copeland link for the relation to inverse Mellin transform. (End)
The n-th row polynomial is (-1/x)^n e^x (x^2*D_x)^n e^(-x). - Tom Copeland, Oct 29 2012
Let f(.,x)^n = f(n,x) = x!/(x-n)!, the falling factorial,and r(.,x)^n = r(n,x) = (x-1+n)!/(x-1)!, the rising factorial, then the Lah polynomials, Lah(n,t)= n!*Sum{k=1..n} binomial(n-1,k-1)(-t)^k/k! (extra sign factor on odd rows), give the transform Lah(n,-f(.,x))= r(n,x), and Lah(n,r(.,x))= (-1)^n * f(n,x). - Tom Copeland, Oct 04 2014
|T(n,k)| = Sum_{j=0..2*(n-k)} A254881(n-k,j)*k^j/(n-k)!. Note that A254883 is constructed analogously from A254882. - Peter Luschny, Feb 10 2015
The T(n,k) are the inverse Bell transform of [1!,2!,3!,...] and |T(n,k)| are the Bell transform of [1!,2!,3!,...]. See A264428 for the definition of the Bell transform and A264429 for the definition of the inverse Bell transform. - Peter Luschny, Dec 20 2015
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates a shifted, signed Narayana matrix A001263. - Tom Copeland, Sep 23 2020

New name using a comment from Wolfdieter Lang by Peter Luschny, May 10 2021

A293604 Expansion of e.g.f.: exp(x * (1 - x)).

Original entry on oeis.org

1, 1, -1, -5, 1, 41, 31, -461, -895, 6481, 22591, -107029, -604031, 1964665, 17669471, -37341149, -567425279, 627491489, 19919950975, -2669742629, -759627879679, -652838174519, 31251532771999, 59976412450835, -1377594095061119, -4256461892701199

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..732

Cf. A000321, A111884, A293571, A293572, A293573.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), this sequence (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-x^2) ))); // G. C. Greubel, Jul 12 2024

CoefficientList[Series[E^(x*(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x))))

a(n) = polhermite(n, 1/2); \\ Michel Marcus, Oct 13 2017

[hermite(n, 1/2) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) = (-1)^n * A000321(n).
a(n) = a(n-1) - 2 * (n-1) * a(n-2) for n > 1.
E.g.f.: Product_{k>=1} (1 + x^k)^(mu(k)/k). - Ilya Gutkovskiy, May 23 2019
a(n) = Hermite(n, 1/2). - G. C. Greubel, Jul 12 2024

A293116 Expansion of e.g.f. exp(x/(x-1)).

Original entry on oeis.org

1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519, 396271, 2442439, 7701409, -145269541, -4833158329, -104056218421, -2002667085119, -37109187217649, -679877731030049, -12440309297451121, -227773259993414719, -4155839606711748061, -74724654677947488521

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..450
Geoffrey B. Campbell, Visible Point Vector Partition Identities for Hyperpyramid Lattices, arXiv:2309.16094 [math.CO], 2023. See pp. 14, 27.

Column k=0 of A293119.

Cf. A000262, A066668, A111884.

a:= proc(n) option remember; `if`(n=0, 1, -add(
      a(n-j)*binomial(n-1, j-1)*j!, j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017

CoefficientList[Series[E^(-x/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(x/(x-1))))

E.g.f.: exp(x/(x-1)).
a(n) = (-1)^n * A111884(n).
E.g.f.: Product_{k>=1} (1 - x^k)^(phi(k)/k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, May 25 2019
D-finite with recurrence a(n) +(-2*n+3)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Mar 13 2023

A066668 Signed row sums of A066667.

Original entry on oeis.org

1, 1, 1, -1, -19, -151, -1091, -7841, -56519, -396271, -2442439, -7701409, 145269541, 4833158329, 104056218421, 2002667085119, 37109187217649, 679877731030049, 12440309297451121, 227773259993414719, 4155839606711748061, 74724654677947488521, 1293162252850914402221

Offset: 0

Christian G. Bower, Dec 17 2001

sign

Numerators in exp(x/(x+1)) power series (signs are different). - Benoit Cloitre, Mar 13 2002

Determinant of n X n matrix M=[m(i,j)] where m(i,i)=i, m(i,j)=1 if i>j, m(i,j)=j-i if j>i. - Vladeta Jovovic, Jan 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A111884.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(x-1))/(1-x)^2)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2018

a := n -> n!*hypergeom([1-n], [2], 1):
seq(simplify(a(n)), n=1..19); # Peter Luschny, Mar 30 2015

CoefficientList[Series[E^(x/(x-1))/(1-x)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2014 *)
Table[Sum[-BellY[n+1, k, -Range[n+1]!], {k, n+1}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))/(1-x)^2)) \\ G. C. Greubel, May 15 2018

A066668 = lambda n: (-1)^n*hypergeometric([-n-1,-n-1,-n],[-n-1],-1)
[Integer(A066668(n).n(100)) for n in range(23)] # Peter Luschny, Sep 22 2014

a(n) = n!LaguerreL(n, 1, 1). - Paul Barry, Sep 08 2004
E.g.f.: exp(x/(x-1))/(1-x)^2.
Conjecture: a(n) +(-2*n+1)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
E.g.f. with a different offset: 1 - product {n >= 1} (1 - x^n)^(phi(n)/n) = x + x^2/2 + x^3/6 - x^4/24 - 19*x^5/120 - ..., where phi(n) = A000010(n) is the Euler totient function. Cf. A000262. - Peter Bala, Jan 01 2014
a(n) = (-1)^n*hypergeom([-n-1,-n-1,-n],[-n-1],-1). - Peter Luschny, Sep 22 2014
a(n) = n!*hypergeom([1-n], [2], 1). - Peter Luschny, Mar 30 2015

A129652 Exponential Riordan array [e^(x/(1-x)),x].

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 73, 52, 18, 4, 1, 501, 365, 130, 30, 5, 1, 4051, 3006, 1095, 260, 45, 6, 1, 37633, 28357, 10521, 2555, 455, 63, 7, 1, 394353, 301064, 113428, 28056, 5110, 728, 84, 8, 1, 4596553, 3549177, 1354788, 340284, 63126, 9198, 1092, 108, 9, 1

Offset: 0

Paul Barry, Apr 26 2007

Satisfies the equation e^[x/(1-x),x] = e*[e^(x/(1-x)),x].
Row sums are A052844.
Antidiagonal sums are A129653.

			Triangle begins:
     1;
     1,    1;
     3,    2,    1;
    13,    9,    3,   1;
    73,   52,   18,   4,  1;
   501,  365,  130,  30,  5, 1;
  4051, 3006, 1095, 260, 45, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7.

Cf. A000262 (column 0), A052844 (row sums).
T(2n,n) gives A350461.
Cf. A052845, A111884, A133314.

A129652 := (n, k) -> (-1)^(k-n+1)*binomial(n,k)*KummerU(k-n+1, 2, -1);
seq(seq(round(evalf(A129652(n,k),99)),k=0..n),n=0..9); # Peter Luschny, Sep 17 2014
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1$2], add((p-> p+
     [0, p[1]*x^j])(b(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=0..n))(b(n)[2]):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 21 2022

T[n_, k_] := If[k==n, 1, n!/k! Sum[Binomial[n-k-1, j]/(j+1)!, {j, 0, n-k-1}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 14 2019 *)

Number triangle T(n,k)=(n!/k!)*sum{i=0..n-k, C(n-k-1,i)/(n-k-i)!}
From Peter Bala, May 14 2012 : (Start)
Array is exp(S*(I-S)^(-1)) where S is A132440 the infinitesimal generator for Pascal's triangle.
Column 0 is A000262.
T(n,k) = binomial(n,k)*A000262(n-k).
So T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then arrange the remaining n-k elements into a set of lists. (End)
T(n,k) = (-1)^(k-n+1)*C(n,k)*KummerU(k-n+1, 2, -1). - Peter Luschny, Sep 17 2014
From Tom Copeland, Mar 11 2016: (Start)
The row polynomials P_n(x) form an Appell sequence with e.g.f. e^(t*P.(x)) = e^[t / (1-t)] e^(x*t), so the lowering and raising operators are L = d/dx = D and the R = x + 1 / (1-D)^2 = x + 1 + 2 D + 3 D^2 + ..., satisfying L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x).
(P.(x) + y)^n = Sum_{k=0..n} binomial(n,k) P_k(x) * y^(n-k) = P_n(x+y).
The Appell polynomial umbral compositional inverse sequence has the e.g.f. e^(t*Q.(x)) = e^[-t / (1-t)] e^(x*t) (see A111884 and A133314), so Q_n(P.(x)) = P_n(Q.(x)) = x^n. The lower triangular matrices for the coefficients of these two Appell sequences are a multiplicative inverse pair.
(End)
Sum_{k=0..n} (-1)^k * T(n,k) = A052845(n). - Alois P. Heinz, Feb 21 2022

A318215 Expansion of e.g.f. exp(x/(1 + x)^2).

Original entry on oeis.org

1, 1, -3, 7, 1, -219, 2581, -22973, 162177, -554039, -10506419, 343049631, -6846400703, 113528248237, -1609627861659, 17371462450651, -36066494745599, -5681921495461743, 243263898097515037, -7398126521141652809, 193119003246643917441, -4476119490014676723659, 89171014860669488040757

Offset: 0

Ilya Gutkovskiy, Aug 21 2018

sign

Cf. A001563, A082579, A111884, A181983.

A318215 := proc(n)
    add((-1)^(n-k)*binomial(n+k-1,2*k-1)*n!/k!,k=0..n) ;
end proc:
seq(A318215(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

nmax = 22; CoefficientList[Series[Exp[x/(1 + x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n + k - 1, 2 k - 1] n!/k!, {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = Sum[(-1)^(k + 1) k k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
Join[{1}, Table[(-1)^(n + 1) n n! HypergeometricPFQ[{1 - n, 1 + n}, {3/2, 2}, 1/4], {n, 22}]]

E.g.f.: Product_{k>=1} exp((-1)^(k+1)*k*x^k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1,2*k-1)*n!/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*k*k!*binomial(n-1,k-1)*a(n-k).
D-finite with recurrence a(n) +(3*n-4)*a(n-1) +(n-1)*(3*n-5)*a(n-2) +(n-1)*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Aug 20 2021

A383991 Series expansion of the exponential generating function exp(-tridend(-x)) - 1 where tridend(x) = (1 - 3x - sqrt(1-6x+x^2)) / (4*x) (A001003).

Original entry on oeis.org

0, 1, -5, 49, -743, 15421, -407909, 13135165, -498874991, 21838772377, -1082819193029, 59983280191561, -3671752681190615, 246130081055714389, -17932045676505509093, 1410893903131294766101, -119227840965746009631839, 10769985399394862863318705

Offset: 0

Michael De Vlieger, May 16 2025

The series -tridend(-x) is the inverse for the substitution of the series trias(x), given by the suspension of the Koszul dual of trias. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..348
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, triassociative operad "Trias".

Cf. A003725, A097388, A111884, A112242, A177885, A318215, A383987, A383990, A383992, A383993, A383994, A383995.

nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[(1 + 3*x - Sqrt[1 + 6*x + x^2])/(4*x)], {x, 0, nn}], x]

A383995 Series expansion of the exponential generating function exp(ff6^!(x)) - 1 where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

Original entry on oeis.org

0, 1, -11, 61, -215, -1559, 62941, -1371131, 26310481, -474554735, 7824076741, -98881279859, -176260664711, 87457412423161, -5077434546358355, 234510433823788501, -10016559114085864799, 413333665704129673249, -16704968283664639137899, 660340818239784197391325

Offset: 0

Michael De Vlieger, May 16 2025

sign

The series ff6^!(x) is the inverse for the substitution of the series ff6(x) (given by A231690), given by the suspension of the Koszul dual of FF6. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..393
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad "FF6".

Cf. A003725, A097388, A111884, A112242, A177885, A318215, A383989, A383990, A383991, A383992, A383993, A383994.

nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[x*(1 - 3*x - x^2 + x^3)/(1 + 3*x + x^2 - x^3)], {x, 0, nn}], x]

A088312 Number of sets of lists (cf. A000262) with even number of lists.

Original entry on oeis.org

1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536, 29668681, 413257790, 6238931821, 101415565836, 1765092183037, 32734873484250, 644215775792401, 13404753632014352, 293976795292186897, 6775966692145553526, 163735077313046119861, 4138498600079573989140

Offset: 0

Vladeta Jovovic, Nov 05 2003

From Peter Bala, Mar 27 2022: (Start)
a(2*n) is odd ; a(2*n+1) is even.
If k is odd then k*(k-1) divides a(k). Consequently, 6 divides a(6*n+3), 10 divides a(10*n+5), 14 divides a(14*n+7), and in general, if k is odd then 2*k divides a(2*k*n + k).
For a positive integer k, a(n+2*k) - a(n) is divisible by k. Thus the sequence obtained by taking a(n) modulo k is purely periodic with period 2*k. Calculation suggests that when k is even the exact period equals k, and when k is odd the exact period equals 2*k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..444
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, LAH transform

Cf. A000262, A001710, A027187, A024429, A024430, A088313, A111884.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Cosh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
A088312 := n -> ifelse(n=0, 1, (1/2)*(n - 1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4)): seq(simplify(A088312(n)), n = 0..21); # Peter Luschny, Dec 14 2022

With[{m=30}, CoefficientList[Series[Cosh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* Vaclav Kotesovec, Jul 04 2015 *)
Table[Sum[n!/(2*k)! Binomial[n - 1, 2*k - 1], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Emanuele Munarini, Sep 03 2017 *)

def A088312_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( cosh(x/(1-x)) ).egf_to_ogf().list()
A088312_list(40) # G. C. Greubel, Dec 13 2022

E.g.f.: cosh(x/(1-x)).
a(n) = Sum_{k=1..floor(n/2)} n!/(2*k)!*binomial(n-1,2*k-1).
a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015
a(n+4) - 2*(2*n+5)*a(n+3) + (6*n^2+24*n+23)*a(n+2) - 2*(n+1)*(n+2)*(2*n+3)*a(n+1) + n*(n+1)^2*(n+2)*a(n) = 0. - Emanuele Munarini, Sep 03 2017
a(n) = (1/2)*(A000262(n) + (-1)^n*A111884(n)). - Peter Bala, Mar 27 2022
a(n) = (1/2)*(n-1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4) for n >= 1. - Peter Luschny, Dec 14 2022

More terms from Vaclav Kotesovec, Jul 04 2015

a(0)-a(1) prepended by Alois P. Heinz, May 10 2016

Showing 1-10 of 28 results. Next

cp's OEIS Frontend

A131202 A coefficient tree from the list partition transform relating A111884, A084358, A000262, A094587, A128229 and A131758.

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A111596 The matrix inverse of the unsigned Lah numbers A271703.

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A293604 Expansion of e.g.f.: exp(x * (1 - x)).

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A293116 Expansion of e.g.f. exp(x/(x-1)).

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A066668 Signed row sums of A066667.

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A129652 Exponential Riordan array [e^(x/(1-x)),x].

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A318215 Expansion of e.g.f. exp(x/(1 + x)^2).

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A383991 Series expansion of the exponential generating function exp(-tridend(-x)) - 1 where tridend(x) = (1 - 3x - sqrt(1-6x+x^2)) / (4*x) (A001003).

A383995 Series expansion of the exponential generating function exp(ff6^!(x)) - 1 where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).