A118791 -id:A118791 - 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)}

A118792 E.g.f.: -log(1 - (1 - sqrt(5 - 4*exp(x)) )/2 ).

Original entry on oeis.org

0, 1, 4, 30, 352, 5670, 116344, 2902830, 85326112, 2887962870, 110620824904, 4730842053630, 223445584599472, 11552029520192070, 648869447924011864, 39347855472366932430, 2562065820090343738432, 178286102174571726213270

Offset: 0

Paul D. Hanna, Apr 30 2006

nonn

Also equals the unsigned row sums of triangle A118791 (offset without leading zero).

			E.g.f.: A(x) = x + 2*x^2 + 30/6*x^3 + 352/24*x^4 + 5670/120*x^5 +...

Cf. A118791.

CoefficientList[Series[-Log[1-(1-Sqrt[5-4*Exp[x]])/2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

a(n)=local(x=X+X^2*O(X^n));n!*polcoeff(-log(1-(1-sqrt(5-4*exp(x)))/2),n,X)

/* As the unsigned row sums of A118791: */ 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,k,X))))

a(n) = (n-1)!*Sum_{k=0..n-1} abs( [x^k] (x/log(1-x-x^2))^n ) for n>0.

a(n) ~ sqrt(5/2)*n^(n-1)/(exp(n)*(log(5/4))^(n-1/2)). - Vaclav Kotesovec, Jun 26 2013

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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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A118792 E.g.f.: -log(1 - (1 - sqrt(5 - 4*exp(x)) )/2 ).

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