A118795 -id:A118795 - cp's OEIS frontend

A118793 Triangle where T(n,k) = -n!/(n-k)!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.

1, -1, 3, 1, -9, 19, -1, 18, -103, 207, 1, -30, 325, -1605, 3211, -1, 45, -785, 6930, -32191, 64383, 1, -63, 1610, -22050, 175861, -790629, 1581259, -1, 84, -2954, 57750, -693861, 5216778, -22974463, 45948927, 1, -108, 4998, -131922, 2213211, -24542910, 177555925, -770820885, 1541641771

Offset: 0

Paul D. Hanna, Apr 30 2006

[0, diagonal] = A052886 with e.g.f.: (1-sqrt(5-4*exp(x)))/2. [0, row sums] = A118794 with e.g.f.: 1-exp((-1+sqrt(5-4*exp(x)))/2). [0, unsigned row sums] = A118795 with e.g.f.: -1+exp((1-sqrt(5-4*exp(x)))/2). Here [0, sequence] indicates that the sequence is to be offset with leading zero.

			Triangle begins:
1;
-1, 3;
1,-9, 19;
-1, 18,-103, 207;
1,-30, 325,-1605, 3211;
-1, 45,-785, 6930,-32191, 64383;
1,-63, 1610,-22050, 175861,-790629, 1581259;
-1, 84,-2954, 57750,-693861, 5216778,-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) + 49/12*x^2 - 5*x^3 + 1439/240*x^4 +...
F(x)^3 = (-1 + 9/2*x - 19/2*x^2) + 15*x^3 - 5161/240*x^4 +...
F(x)^4 = ( 1 - 18/3*x + 103/6*x^2 - 207/6*x^3) + 42239/720*x^4 +...
F(x)^5 = (-1 + 30/4*x - 325/12*x^2 + 1605/24*x^3 - 3211/24*x^4) +...

Cf. A052886 (diagonal), A118794 (row sums), A118795 (unsigned row sums); A118791 (variant).

{T(n,k)=local(x=X+X^2*O(X^(k+2)));-n!/(n-k)!*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

Paul D. Hanna, Apr 30 2006

nonn

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

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

Cf. A118793, A118795.

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 *)

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 */

{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),", "))

/* 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),", "))

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

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A118793 Triangle where T(n,k) = -n!/(n-k)!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.

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

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