A144006 Triangle, read by rows of coefficients of x^n*y^k for k=0..n(n-1)/2 for n>=0, defined by e.g.f.: A(x,y) = 1 + Series_Reversion( Integral A(-x*y,y) dx ), with leading zeros in each row suppressed.

1, 1, 1, 3, -1, 15, -10, 3, -1, 105, -105, 55, -30, 10, -3, 1, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1, 135135, -270270, 294525, -275275, 228375

Offset

0,4

Links

Table of n, a(n) for n=0..47.

Formula

E.g.f. satisfies: A(x,y) = 1 + Series_Reversion[Integral A(-x*y,y) dx].

T(n,k) = [x^n*y^k] n!*A(x,y) for k=0..n(n-1)/2, n>=0.

Row sums equal A144005.

A067146(n) = Sum_{k=0..n(n-1)/2} (-1)^k*T(n,k).

Conjectured to be a signed version of table A014621.

Conjecture: row sums form A014623.

Conjecture: unsigned row sums form A014622.

Example

Triangle begins (without suppressing leading zeros):
1;
1;
0, 1;
0,0, 3, -1;
0,0,0, 15, -10, 3, -1;
0,0,0,0, 105, -105, 55, -30, 10, -3, 1;
0,0,0,0,0, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1;
0,0,0,0,0,0, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1;
0,0,0,0,0,0,0, 135135, -270270, 294525, -275275, 228375, -172200, 120960, -78519, 48006, -28336, 16065, -8609, 4461, -2166, 1018, -470, 196, -76, 30, -10, 3, -1; ...

Prog

(PARI) {T(n, k)=local(A=1+x*O(x^n)); for(i=0, n, A=1+serreverse(intformal(subst(A, x, -x*y)))); n!*polcoeff(polcoeff(A, n, x), k, y)}

Crossrefs

Cf. A144005, A067146.

Conjectured to generate A014621, A014622 and A014623, which are related to Levine's sequence A011784.

Sequence in context: A134685 A130757 A181996 * A014621 A193966 A366120

Adjacent sequences: A144003 A144004 A144005 * A144007 A144008 A144009

Keyword

sign,tabf

Author

Paul D. Hanna, Sep 10 2008

Status

approved