A136590 -id:A136590 - cp's OEIS frontend

A136595 Matrix inverse of triangle A136590.

1, 0, 1, 0, -1, 1, 0, 7, -3, 1, 0, -61, 31, -6, 1, 0, 751, -375, 85, -10, 1, 0, -11821, 5911, -1350, 185, -15, 1, 0, 226927, -113463, 26341, -3710, 350, -21, 1, 0, -5142061, 2571031, -603246, 87381, -8610, 602, -28, 1, 0, 134341711, -67170855, 15887845, -2346330, 240051, -17766

Offset: 0

Paul D. Hanna, Jan 10 2008

A136590 is the triangle of trinomial logarithmic coefficients.
Column 1 is signed A048287, which is the number of semiorders on n labeled nodes whose incomparability graph is connected.
The Bell transform of (-1)^n*A048287(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins:
1;
0, 1;
0, -1, 1;
0, 7, -3, 1;
0, -61, 31, -6, 1;
0, 751, -375, 85, -10, 1;
0, -11821, 5911, -1350, 185, -15, 1;
0, 226927, -113463, 26341, -3710, 350, -21, 1;
0, -5142061, 2571031, -603246, 87381, -8610, 602, -28, 1;
0, 134341711, -67170855, 15887845, -2346330, 240051, -17766, 966, -36, 1; ...

Cf. columns: A048287, A136596, A136597; A136590 (matrix inverse); A136588, A136589.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^n*A048287(n+1), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 11;
M = BellMatrix[Sum[(-1)^(k+1) k! StirlingS2[#+1, k] CatalanNumber[k-1], {k, 1, #+1}]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

PARI
```
{T(n,k) = if(n
				
```

/* Define Stirling2: */
{Stirling2(n,k) = n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!,n)}
/* Define Catalan(m,n) = [x^n] C(x)^m: */
{CATALAN(m,n) = binomial(2*n+m,n) * m/(2*n+m)}
/* Define this triangle: */
{T(n,k) = if(n

# uses[bell_matrix from A264428]
bell_matrix(lambda n: (-1)^n*A048287(n+1), 10) # Peter Luschny, Jan 18 2016

T(n,k) = Sum_{i=0..n-1} (-1)^i * (k+i)! * Stirling2(n,k+i) * Catalan(k,i)/k!, where Stirling2(n,k) = A008277(n,k); Catalan(k,i) = C(2i+k,i)*k/(2i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).

A136591 Column 1 of triangle A136590.

Original entry on oeis.org

1, 1, -4, 6, 24, -240, 720, 5040, -80640, 362880, 3628800, -79833600, 479001600, 6227020800, -174356582400, 1307674368000, 20922789888000, -711374856192000, 6402373705728000, 121645100408832000, -4865804016353280000, 51090942171709440000

Offset: 1

Paul D. Hanna, Jan 10 2008

sign

Cf. A136590, A136592, A136593, A136594; A061347.

{a(n) = n!*polcoeff( log(1+x+x^2 +x*O(x^(n+1))),n)}
for(n=1,30,print1(a(n),", "))

E.g.f.: log(1 + x + x^2).

a(n) = n!*A061347(n) for n>=1.

A136592 Column 2 of triangle A136590.

Original entry on oeis.org

1, 3, -13, -10, 394, -2016, -5076, 170064, -1155024, -5005440, 193724640, -1656720000, -10280355840, 465087087360, -4804977542400, -39012996556800, 2035558551398400, -24660231399014400, -248246498826547200, 14713557956794368000

Offset: 2

Paul D. Hanna, Jan 10 2008

sign

			E.g.f.: A(x) = 1*x^2/2! + 3*x^3/3! - 13*x^4/4! - 10*x^5/5! + 394*x^6/6! +...

Cf. A136590, A136591, A136593, A136594.

With[{nn=30},Drop[CoefficientList[Series[Log[1+x+x^2]^2/2,{x,0,nn}],x] Range[ 0,nn]!,2]] (* Harvey P. Dale, Feb 28 2013 *)

a(n)=(n+2)!*polcoeff(log(1+x+x^2 +x*O(x^(n+2)))^2/2!,n+2)

E.g.f.: A(x) = log(1 + x + x^2)^2 / 2!.

A136593 Column 3 of triangle A136590.

Original entry on oeis.org

1, 6, -25, -135, 1834, -3668, -110692, 1339020, -1181664, -164709864, 2206092096, 395662176, -463716547776, 7029335571840, 8900411569920, -2265668505227520, 38689597829053440, 92447263589921280, -17785648201625856000, 338957966532455424000

Offset: 3

Paul D. Hanna, Jan 10 2008

sign

			E.g.f.: A(x) = 1*x^3/3! + 6*x^4/4! - 25*x^5/5! - 135*x^6/6! + 1834*x^7/7! -...

Cf. A136590, A136591, A136592, A136594.

a(n)=(n+3)!*polcoeff(log(1+x+x^2 +x*O(x^(n+3)))^3/3!,n+3)

E.g.f.: A(x) = log(1 + x + x^2)^3 / 3!.

A136594 Unsigned row sums of triangle A136590.

Original entry on oeis.org

1, 1, 2, 8, 26, 70, 820, 5152, 20316, 388712, 3666188, 17298908, 501805832, 6256792412, 33844737292, 1353617016078, 20960708128068, 137741948419428, 6588092586831028, 121860622573650906, 924837580461274556

Offset: 0

Paul D. Hanna, Jan 10 2008

nonn

A136590 is the triangle of trinomial logarithmic coefficients.

Cf. A136590, A136591, A136592, A136593.

a(n)=sum(k=0,n,n!/k!*abs(polcoeff(log(1+x+x^2 +x*O(x^n))^k,n)))

A048287 Number of semiorders on n labeled nodes whose incomparability graph is connected.

Original entry on oeis.org

1, 1, 7, 61, 751, 11821, 226927, 5142061, 134341711, 3975839341, 131463171247, 4803293266861, 192178106208271, 8356430510670061, 392386967808249967, 19788154572706556461, 1066668756919315412431, 61204224384073232815981

Offset: 1

Richard Stanley

			a(3)=7, the seven semiorders being three disjoint points and the disjoint union of a point and a two-element chain (with six labelings).

Robert Israel, Table of n, a(n) for n = 1..373
Mats Granvik, Power series to calculate Lambert W up to infinity

Cf. A000108, A006531.

Cf. A136595, A136590.

A048287 := n -> add((-1)^(n-k-1)*Stirling2(n,k+1)*(2*k)!/k!, k=0..n-1):
seq(A048287(n), n=1..18); # Peter Luschny, Jan 27 2016

Table[Sum[(-1)^(n - k) StirlingS2[n, k] k!*CatalanNumber[k - 1], {k, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)
Rest[Range[0, 18]! CoefficientList[Series[1 - 2 (1 - Exp[-x]) /(1 - Sqrt[4 Exp[-x] - 3]), {x, 0, 18}], x]] (* Vincenzo Librandi, Jan 28 2016 *)

{a(n)=local(A136590=matrix(n+1,n+1,r,c,if(r>=c,(r-1)!/(c-1)!*polcoeff(log(1+x+x^2 +x*O(x^n))^(c-1),r-1))));(-1)^(n+1)*(A136590^-1)[n+1,2]} \\ Paul D. Hanna, Jan 10 2008

{a(n) = if( n<0, 0, n! * polcoeff( (1 - sqrt(4*exp(-x + x*O(x^n)) - 3)) / 2, n))}; /* Michael Somos, Nov 26 2017 */

{a(n) = if( n<1, 0, n! * polcoeff( serreverse( -log(1 - x + x^2 + x * O(x^n))), n))}; /* Michael Somos, Nov 26 2017 */

E.g.f.: 1-2*(1-exp(-x))/(1-sqrt(4*exp(-x)-3)).
E.g.f.: (1 - sqrt(4*exp(-x) - 3)) / 2. - Michael Somos, Nov 26 2017
a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n, k)*k!*Catalan(k-1). - Vladeta Jovovic, Oct 18 2003
Equals column 1 (unsigned) of triangle A136595, which is the matrix inverse of the triangle A136590 of trinomial logarithmic coefficients. - Paul D. Hanna, Jan 10 2008
E.g.f A(x)=F(exp(x)-1), F(x)=x*A005043(x). - Vladimir Kruchinin, Sep 07 2010
a(n) = (-1)^(n-1) + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - Robert Israel, Mar 01 2016
Given e.g.f. =: A(x), then exp(-x) = A(x)^2 - A(x) + 1 = A'(x)*(1 - 2*A(x)). - Michael Somos, Nov 26 2017
a(n) ~ sqrt(3/8) * n^(n-1) / (log(4/3)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Dec 16 2020

More terms from Vladeta Jovovic, Oct 18 2003

cp's OEIS Frontend

A136595 Matrix inverse of triangle A136590.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Sage

Formula

A136591 Column 1 of triangle A136590.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A136592 Column 2 of triangle A136590.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A136593 Column 3 of triangle A136590.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A136594 Unsigned row sums of triangle A136590.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A048287 Number of semiorders on n labeled nodes whose incomparability graph is connected.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions