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A109956 Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.

%I A109956 #28 May 26 2025 09:52:53
%S A109956 1,-1,1,3,-4,1,-12,18,-7,1,55,-88,42,-10,1,-273,455,-245,75,-13,1,
%T A109956 1428,-2448,1428,-510,117,-16,1,-7752,13566,-8379,3325,-910,168,-19,1,
%U A109956 43263,-76912,49588,-21252,6578,-1472,228,-22,1,-246675,444015,-296010,134550,-45630,11700,-2223,297,-25,1
%N A109956 Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.
%C A109956 Riordan array (g,f) where f/(1-f)^3=x and g=1-f.
%C A109956 First column is (-1)^n*binomial(3n,n)/(2n+1), a signed version of A001764.
%C A109956 Second column is a signed version of A006629.
%C A109956 Diagonal sums are A109957.
%H A109956 Paul Barry, <a href="https://arxiv.org/abs/2505.16718">d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths</a>, arXiv:2505.16718 [math.CO], 2025. See p. 13.
%H A109956 Paul Drube, <a href="https://arxiv.org/abs/2007.01892">Generalized Path Pairs and Fuss-Catalan Triangles</a>, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8 (up to signs).
%F A109956 Number triangle T(n, k) = (-1)^(n-k)*((3k+1)/(2n+k+1))*binomial(3n, n-k).
%F A109956 From _Werner Schulte_, Oct 27 2015: (Start)
%F A109956 If u(m,n) = (-1)^n*(Sum_{k=0..n} T(n,k)*((m+1)*k+1)) and v(m,n) = (-1)^n*(Sum_{k=0..n} (-1)^k*T(n,k)*m^k) and D(x) is the g.f. of A001764 then P(m,x) = Sum_{n>=0} u(m,n)*x^n = 1-(m+1)*x*D(x)^2 and Q(m,x) = Sum_{n>=0} v(m,n)*x^n = 1/P(m,x).
%F A109956 If G(k,x) is the g.f. of column k (k>=0) then G(k,x) = G(0,x)^(3*k+1). (End)
%e A109956 Triangle begins:
%e A109956       1;
%e A109956      -1,   1;
%e A109956       3,  -4,    1;
%e A109956     -12,  18,   -7,   1;
%e A109956      55, -88,   42, -10,   1;
%e A109956    -273, 455, -245,  75, -13, 1;
%e A109956    ...
%p A109956 # Function RiordanSquare defined in A321620.
%p A109956 tt := sin(arcsin(3*sqrt(x*3/4))/3)/sqrt(x*3/4): R := RiordanSquare(tt, 11):
%p A109956 seq(seq(LinearAlgebra:-Row(R,n)[k]*(-1)^(n+k), k=1..n), n=1..11); # _Peter Luschny_, Nov 27 2018
%t A109956 T[n_, k_] := (-1)^(n - k)((3k + 1)/(2n + k + 1)) Binomial[3n, n - k];
%t A109956 Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* _Jean-François Alcover_, Jun 13 2019 *)
%o A109956 (PARI) tabl(nn) = {my(m = matrix(nn, nn, n, k, if (n<k, 0, binomial(n+2*k-3, 3*k-3)))); m = 1/m; for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ _Michel Marcus_, Nov 20 2015
%Y A109956 Cf. A001764, A109955, A109957, A321620.
%K A109956 easy,sign,tabl
%O A109956 0,4
%A A109956 _Paul Barry_, Jul 06 2005