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

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

0,3

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.

Links

Table of n, a(n) for n=0..40.

Example

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! +...

Prog

(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)}

Crossrefs

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

Sequence in context: A154343 A049969 A088634 * A234840 A234743 A284989

Adjacent sequences: A118788 A118789 A118790 * A118792 A118793 A118794

Keyword

sign,tabl

Author

Paul D. Hanna, Apr 30 2006

Status

approved