cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A136596 Column 2 of triangle A136595.

Original entry on oeis.org

1, -3, 31, -375, 5911, -113463, 2571031, -67170855, 1987919671, -65731585623, 2401646633431, -96089053104135, 4178215255335031, -196193483904124983, 9894077286353278231, -533334378459657706215, 30602112192036616407991
Offset: 2

Views

Author

Paul D. Hanna, Jan 10 2008

Keywords

Crossrefs

Cf. A136595; A048287, A136597.

Programs

  • PARI
    {a(n)=n!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(2+i)),n)*binomial(2*i+2,i)/(2*i+2))}
for(n=2,20,print1(a(n),", "))
  • PARI
    /* 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 sequence: */
{a(n)=sum(i=0,n-1,(-1)^i*(2+i)!*Stirling2(n,2+i)*Catalan(2,i)/2!)}
for(n=2,20,print1(a(n),", "))

Formula

a(n) = Sum_{i=0..n-1} (-1)^i*(2+i)!*Stirling2(n,2+i)*Catalan(2,i)/2!, where Stirling2(n,k) = A008277(n,k); Catalan(k,i) = binomial(2*i+k,i)*k/(2*i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).
a(n) = (1+(-1)^n*A048287(n))/2. - Vladeta Jovovic, Jan 27 2008

A136597 Column 3 of triangle A136595.

Original entry on oeis.org

1, -6, 85, -1350, 26341, -603246, 15887845, -473148150, 15723174181, -576826897086, 23157022930405, -1009818279438150, 47533643556874021, -2402218856253008526, 129730266330534913765, -7455932648513351731350, 454377365410347843373861
Offset: 3

Views

Author

Paul D. Hanna, Jan 10 2008

Keywords

Crossrefs

Cf. A136595, A048287, A136596.

Programs

  • PARI
    a(n)=n!/2!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(3+i)),n)*binomial(2*i+3,i)/(2*i+3))
  • PARI
    /* 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 sequence: */ {a(n)=sum(i=0,n-1,(-1)^i*(3+i)!*Stirling2(n,3+i)*Catalan(3,i)/3!)}

Formula

a(n) = Sum_{i=0..n-1} (-1)^i*(3+i)!*Stirling2(n,3+i)*Catalan(3,i)/3!, where Stirling2(n,k) = A008277(n,k), Catalan(k,i) = C(2*i+k,i)*k/(2*i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).

A136588 a(n) = Sum_{k=0..n} A136595(n,k)*n^k.

Original entry on oeis.org

1, 1, 2, 21, 124, 1880, 20046, 391419, 6195288, 147481299, 3121373690, 87790122816, 2329580861268, 75790954533385, 2415630777959686, 89478235732836705, 3323789119614522416, 138402773923330655700
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2008

Keywords

Crossrefs

Cf. A136595; A048287, A136596, A136597; A136589.

Programs

  • PARI
    {a(n)=sum(k=0,n,if(k==0,0^n,n^k*n!/(k-1)!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(k+i)),n)*binomial(2*i+k,i)/(2*i+k))))}

A136589 a(n) = Sum_{k=0..n} A136595(n,k)*(n+1)^k.

Original entry on oeis.org

1, 2, 6, 44, 345, 4182, 53571, 905144, 16154550, 349976610, 8015743483, 211987872516, 5925745900685, 185188128471374, 6108604964499810, 220403394009702384, 8375181440031684305, 342816825221831030490
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2008

Keywords

Crossrefs

Cf. A136595; A048287, A136596, A136597; A136588.

Programs

  • PARI
    {a(n)=sum(k=0,n,if(k==0,0^n,(n+1)^k*n!/(k-1)!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(k+i)),n)*binomial(2*i+k,i)/(2*i+k))))}

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

Views

Author

Richard Stanley

Keywords

Examples

			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).

Links

Crossrefs

Cf. A000108, A006531.
Cf. A136595, A136590.

Programs

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

Formula

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

Extensions

More terms from Vladeta Jovovic, Oct 18 2003

A136590 Triangle of trinomial logarithmic coefficients: A027907(n,k) = Sum_{i=0..k} T(k,i)*n^i/k!.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -4, 3, 1, 0, 6, -13, 6, 1, 0, 24, -10, -25, 10, 1, 0, -240, 394, -135, -35, 15, 1, 0, 720, -2016, 1834, -525, -35, 21, 1, 0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1, 0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1, 0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2008

Keywords

Comments

A027907 is the triangle of trinomial coefficients.
The Bell transform of A136591(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			Triangle begins:
1;
0, 1;
0, 1, 1;
0, -4, 3, 1;
0, 6, -13, 6, 1;
0, 24, -10, -25, 10, 1;
0, -240, 394, -135, -35, 15, 1;
0, 720, -2016, 1834, -525, -35, 21, 1;
0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1;
0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1;
0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1; ...
Trinomial coefficients can be calculated as illustrated by:
A027907(4,3) = (T(3,0)*4^0 + T(3,1)*4^1 + T(3,2)*4^2 + T(3,3)*4^3)/3! =
(0 - 4*4 + 3*4^2 + 1*4^3)/3! = 96/6 = 16.

Crossrefs

Cf. columns: A136591, A136592, A136593; A136594 (unsigned row sums); A136595 (matrix inverse); A027907, A002426.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> n!*(modp(n+1,3)-modp(n,3)), 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    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[#!*(Mod[# + 1, 3] - Mod[#, 3])&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • PARI
    {T(n,k)=n!/k!*polcoeff(log(1+x+x^2 +x*O(x^n))^k,n)}
for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))
  • Sage
    # uses[bell_matrix from A264428]
bell_matrix(lambda n: A136591(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

E.g.f. of column k = log(1 + x + x^2)^k / k! for k>=0.
Central trinomial coefficients: A002426(n) = Sum_{k=0..n} T(n,k)*n^k/n!.
Showing 1-6 of 6 results.

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