A118794 -id:A118794 - 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)}

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

Paul D. Hanna, Apr 30 2006

nonn

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

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

Cf. A118793, A118794.

CoefficientList[Series[-1 + E^((1-Sqrt[5-4*E^x])/2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 14 2014 *)

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 */

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

/* 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))))

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

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

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