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%I #17 Jul 19 2024 22:25:01
%S 1,1,1,3,-1,15,-10,3,-1,105,-105,55,-30,10,-3,1,945,-1260,910,-630,
%T 350,-168,76,-30,10,-3,1,10395,-17325,15750,-12880,9135,-5789,3381,
%U -1806,910,-434,196,-76,30,-10,3,-1,135135,-270270,294525,-275275,228375
%N 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 E.g.f. satisfies: A(x,y) = 1 + Series_Reversion[Integral A(-x*y,y) dx].
%F T(n,k) = [x^n*y^k] n!*A(x,y) for k=0..n(n-1)/2, n>=0.
%F Row sums equal A144005.
%F A067146(n) = Sum_{k=0..n(n-1)/2} (-1)^k*T(n,k).
%F This is a signed version of table A014621 because setting f((1+x)/y):=A(-x*y,y)/y for fixed y>0 implies f(f(x))*f'(x)=-1 and f(1/y)=1/y, as in the second formula of A014621. Therefore, the row sums form A014623 and the unsigned row sums form A014622. - _Roland Miyamoto_, Jun 03 2024
%e Triangle begins (without suppressing leading zeros):
%e 1;
%e 1;
%e 0, 1;
%e 0,0, 3, -1;
%e 0,0,0, 15, -10, 3, -1;
%e 0,0,0,0, 105, -105, 55, -30, 10, -3, 1;
%e 0,0,0,0,0, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1;
%e 0,0,0,0,0,0, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1;
%e 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 (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 Cf. A144005, A067146.
%Y Generates A014621, A014622 and A014623, which are related to Levine's sequence A011784.
%K sign,tabf
%O 0,4
%A _Paul D. Hanna_, Sep 10 2008
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