This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A109954 #15 Mar 15 2025 23:29:09
%S A109954 1,-3,1,6,-5,1,-10,15,-7,1,15,-35,28,-9,1,-21,70,-84,45,-11,1,28,-126,
%T A109954 210,-165,66,-13,1,-36,210,-462,495,-286,91,-15,1,45,-330,924,-1287,
%U A109954 1001,-455,120,-17,1,-55,495,-1716,3003,-3003,1820,-680,153,-19,1,66,-715,3003,-6435,8008,-6188,3060,-969,190,-21,1
%N A109954 Riordan array (1/(1+x)^3,x/(1+x)^2).
%C A109954 Inverse of Riordan array (c(x)^3,x*c(x)^2) or A050155, with c(x) the g.f. of A000108. Unsigned array is the Riordan array (1/(1-x)^3,x/(1-x)^2), with T(n,k) = binomial(n+k+2,2*k+2).
%C A109954 Triangle of coefficients of polynomials defined by: c0=1; p(x, n) = (2 + c0 - x)*p(x, n - 1) + (-1 - c0 (2 - x))*p(x, n - 2) + c0*p(x, n - 3). Setting c0=0 gives A136674. - _Roger L. Bagula_, Apr 08 2008
%C A109954 The triangle entries Ts(n,k):=(-1)^(n-1)*A109954(n-1, k) = ((-1)^k)*binomial(n+k+1, 2(k+1)), n>=1, k=0..n-1, are the coefficients of x^(2*k) of the polynomial P(n,x^2) := (1 - (-1)^n*S(2*n,x))/x^2, with the Chebyshev S-polynomials with coefficient triangle given in A049310.
%C A109954 P(n,x^2) = - R(n+1,x)*S(n-1,x)/x^2 if n is even and P(n,x^2) = R(n,x)*S(n,x)/x^2 if n is odd, with R the monic integer Chebyshev T-polynomials with coefficient triangle given in A127672. - _Wolfdieter Lang_, Oct 24 2012.
%F A109954 Number triangle T(n, k) = (-1)^(n+k)*binomial(n+k+2, 2*k+2) [offset (0, 0)].
%e A109954 Triangle T(n, k) begins:
%e A109954 n/k 0 1 2 3 4 5 6 7 8 9 10
%e A109954 0: 1
%e A109954 1: -3 1
%e A109954 2: 6 -5 1
%e A109954 3: -10 15 -7 1
%e A109954 4: 15 -35 28 -9 1
%e A109954 5: -21 70 -84 45 -11 1
%e A109954 6: 28 -126 210 -165 66 -13 1
%e A109954 7: -36 210 -462 495 -286 91 -15 1
%e A109954 8: 45 -330 924 -1287 1001 -455 120 -17 1
%e A109954 9: -55 495 -1716 3003 -3003 1820 -680 153 -19 1
%e A109954 10: 66 -715 3003 -6435 8008 -6188 3060 -969 190 -21 1
%e A109954 ... Reformatted and extended by Wolfdieter Lang, Oct 24 2012.
%t A109954 c0 = 1; p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = 2 - x + c0; p[x_, n_] := p[x, n] = (2 + c0 - x)*p[x, n - 1] + (-1 - c0 (2 - x))*p[x, n - 2] + c0*p[x, n - 3]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] - _Roger L. Bagula_, Apr 08 2008
%Y A109954 Cf. A129818, A085478.
%K A109954 easy,sign,tabl
%O A109954 0,2
%A A109954 _Paul Barry_, Jul 06 2005