Search: id:A144006
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%I A144006
%S A144006 1,1,1,3,1,15,10,3,1,105,105,55,30,10,3,1,945,1260,910,630,350,
%T A144006 168,76,30,10,3,1,10395,17325,15750,12880,9135,5789,3381,1806,
%U A144006 910,434,196,76,30,10,3,1,135135,270270,294525,275275,228375
%V A144006 1,1,1,3,-1,15,-10,3,-1,105,-105,55,-30,10,-3,1,945,-1260,910,-630,350,
%W A144006 -168,76,-30,10,-3,1,10395,-17325,15750,-12880,9135,-5789,3381,-1806,
%X A144006 910,-434,196,-76,30,-10,3,-1,135135,-270270,294525,-275275,228375
%N 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.
%F A144006 E.g.f. satisfies: A(x,y) = 1 + Series_Reversion[Integral A(-x*y,y) dx].
%F A144006 T(n,k) = [x^n*y^k] n!*A(x,y) for k=0..n(n-1)/2, n>=0.
%F A144006 Row sums equal A144005.
%F A144006 A067146(n) = Sum_{k=0..n(n-1)/2} (-1)^k*T(n,k).
%F A144006 Conjectured to be a signed version of table A014621.
%F A144006 Conjecture: row sums form A014623.
%F A144006 Conjecture: unsigned row sums form A014622.
%e A144006 Triangle begins (without suppressing leading zeros):
%e A144006 1;
%e A144006 1;
%e A144006 0, 1;
%e A144006 0,0, 3, -1;
%e A144006 0,0,0, 15, -10, 3, -1;
%e A144006 0,0,0,0, 105, -105, 55, -30, 10, -3, 1;
%e A144006 0,0,0,0,0, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1;
%e A144006 0,0,0,0,0,0, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 
               910, -434, 196, -76, 30, -10, 3, -1;
%e A144006 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; ...
%o A144006 (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)}
%Y A144006 Cf. A144005, A067146.
%Y A144006 Conjectured to generate A014621, A014622 and A014623, which are related 
               to Levine's sequence A011784.
%Y A144006 Sequence in context: A134685 A130757 A014621 this_sequence A113378 A156289 
               A095922
%Y A144006 Adjacent sequences: A144003 A144004 A144005 this_sequence A144007 A144008 
               A144009
%K A144006 sign,tabf
%O A144006 0,4
%A A144006 Paul D. Hanna (pauldhanna(AT)juno.com), Sep 10 2008

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