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 1-3 of 3 results.

A118791 Triangle where T(n,k) = -n!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.

Original entry on oeis.org

1, -1, 3, 2, -9, 19, -6, 36, -103, 207, 24, -180, 650, -1605, 3211, -120, 1080, -4710, 13860, -32191, 64383, 720, -7560, 38640, -132300, 351722, -790629, 1581259, -5040, 60480, -354480, 1386000, -4163166, 10433556, -22974463, 45948927, 40320, -544320, 3598560, -15830640, 53117064
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2006

Keywords

Comments

[0, diagonal] = A052886 with e.g.f.: (1-sqrt(5-4*exp(x)))/2. [0, row sums] = A117271 with e.g.f.: log((3-sqrt(5-4*exp(x)))/2). [0, unsigned row sums] = A118792 with e.g.f.: -log((1+sqrt(5-4*exp(x)))/2). Here [0, sequence] indicates that the sequence is offset with a leading zero.

Examples

			Triangle begins:
1;
-1, 3;
2,-9, 19;
-6, 36,-103, 207;
24,-180, 650,-1605, 3211;
-120, 1080,-4710, 13860,-32191, 64383;
720,-7560, 38640,-132300, 351722,-790629, 1581259;
-5040, 60480,-354480, 1386000,-4163166, 10433556,-22974463, 45948927;
which is formed from the powers of F(x) = x/log(1-x-x^2):
F(x)^1 = (-1) + 3/2*x - 11/12*x^2 + 9/8*x^3 - 641/720*x^4 +...
F(x)^2 = ( 1 - 3*x)/1! + 49/12*x^2 - 5*x^3 + 1439/240*x^4 +...
F(x)^3 = (-2 + 9*x - 19*x^2)/2! + 15*x^3 - 5161/240*x^4 +...
F(x)^4 = ( 6 - 36*x + 103*x^2 - 207*x^3)/3! + 42239/720*x^4 +...
F(x)^5 = (-24 + 180*x - 650*x^2 + 1605*x^3 - 3211*x^4)/4! +...

Crossrefs

Cf. A052886 (diagonal), A117271 (row sums), A118792 (unsigned row sums); A118793 (variant).

Programs

  • PARI
    {T(n,k)=local(x=X+X^2*O(X^(k+2)));-n!*polcoeff(((x/log(1-x-x^2)))^(n+1),k,X)}

A118794 E.g.f.: 1 - exp((-1 + sqrt(5 - 4*exp(x)) )/2).

Original entry on oeis.org

0, 1, 2, 11, 121, 1902, 38381, 945989, 27552260, 925920081, 35265751869, 1501219998148, 70632987480771, 3639861179067661, 203881981765871618, 12333901891547136559, 801418950244634922973, 55665376886060309513990
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2006

Keywords

Comments

Also equals the row sums of triangle A118793 (offset without leading zero).

Examples

			E.g.f.: A(x) = x + 2/2*x^2 + 11/6*x^3 + 121/24*x^4 + 1902/120*x^5 + ...

Crossrefs

Cf. A118793, A118795.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1-Exp[(Sqrt[5-4Exp[x]]-1)/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 07 2014 *)
  • Maxima
    a(n):=sum(((-1)^(m+1)*sum(((k+m-1)!*binomial(2*k+m-1,k+m-1)*stirling2(n,k+m)),k,0,n-m))/(m-1)!,m,1,n); /* Vladimir Kruchinin, Jul 02 2011 */
  • PARI
    {a(n)=local(x=X+X^2*O(X^n));n!*polcoeff(1-exp((-1+sqrt(5-4*exp(x)))/2),n,X)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* As the row sums of A118793: */
{a(n)=local(x=X+X^2*O(X^n));if(n<1,0, -(n-1)!*sum(k=0,n-1,polcoeff(((x/log(1-x-x^2)))^n/(n-1-k)!,k,X)))}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = (n-1)!*Sum_{k=0..n-1} [x^k] (x/log(1-x-x^2))^n/(n-1-k)! for n>0.
a(n) = sum(m=1..n, ((-1)^(m+1)*sum(k=0..n-m, ((k+m-1)!*binomial(2*k+m-1,k+m-1)*stirling2(n,k+m))))/(m-1)!). - Vladimir Kruchinin, Jul 02 2011
a(n) ~ sqrt(5/2) * n^(n-1) / (2 * exp(n+1/2) * (log(5/4))^(n-1/2)). - Vaclav Kotesovec, Jul 31 2014

A118795 E.g.f.: -1 + exp(( 1 - sqrt(5 - 4*exp(x)) )/2).

Original entry on oeis.org

0, 1, 4, 29, 329, 5172, 104335, 2571473, 74894818, 2516911731, 95862252417, 4080739041238, 192000366357981, 9894168501171229, 554208686184384028, 33527021385789228265, 2178482569432714859789, 151314182463701892157460, 11188187745418763137485747
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2006

Keywords

Comments

Also equals the unsigned row sums of triangle A118793 (offset without leading zero).

Examples

			E.g.f.: A(x) = x + (4/2)*x^2 + (29/6)*x^3 + (329/24)*x^4 + (5172/120)*x^5 + ...

Crossrefs

Cf. A118793, A118794.

Programs

  • Mathematica
    CoefficientList[Series[-1 + E^((1-Sqrt[5-4*E^x])/2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 14 2014 *)
  • Maxima
    a(n):=sum((sum(((i+k-1)!*binomial(k+2*i-1,i+k-1)*stirling2(n,i+k)), i,0,n-k))/(k-1)!,k,1,n); /* Vladimir Kruchinin, Nov 22 2011 */
  • PARI
    a(n)=local(x=X+X^2*O(X^n));n!*polcoeff(-1+exp((1-sqrt(5-4*exp(x)))/2),n,X)
  • PARI
    /* As the unsigned row sums of A118793: */ a(n)=local(x=X+X^2*O(X^n));if(n<1,0, (n-1)!*sum(k=0,n-1,abs(polcoeff(((x/log(1-x-x^2)))^n/(n-1-k)!,k,X))))

Formula

a(n) = (n-1)!*Sum_{k=0..n-1} abs( [x^k] (x/log(1-x-x^2))^n/(n-1-k)! ) for n>0.
a(n) = sum(k=1..n, (sum(i=0..n-k, ((i+k-1)!*C(k+2*i-1,i+k-1) *stirling2(n, i+k))))/(k-1)!). - Vladimir Kruchinin, Nov 22 2011
a(n) ~ sqrt(5) * n^(n-1) / (2^(3/2) * exp(n-1/2) * (log(5/4))^(n-1/2)). - Vaclav Kotesovec, Jul 14 2014
Showing 1-3 of 3 results.

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