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

0,3

Comments

[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.

Links

Table of n, a(n) for n=0..44.

Example

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) +...

Prog

(PARI) {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)}

Crossrefs

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

Sequence in context: A111568 A209324 A121489 * A247231 A160568 A157403

Adjacent sequences: A118790 A118791 A118792 * A118794 A118795 A118796

Keyword

sign,tabl

Author

Paul D. Hanna, Apr 30 2006

Status

approved