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A283151 Triangle read by rows: Riordan array (1/(1-9x)^(2/3), x/(9x-1)).

%I A283151 #45 Jan 05 2025 19:51:41
%S A283151 1,6,-1,45,-15,1,360,-180,24,-1,2970,-1980,396,-33,1,24948,-20790,
%T A283151 5544,-693,42,-1,212058,-212058,70686,-11781,1071,-51,1,1817640,
%U A283151 -2120580,848232,-176715,21420,-1530,60,-1,15677145,-20902860,9754668,-2438667,369495,-35190,2070,-69,1,135868590,-203802885
%N A283151 Triangle read by rows: Riordan array (1/(1-9x)^(2/3), x/(9x-1)).
%C A283151 This is an example of a Riordan group involution.
%C A283151 Dual Riordan array of A283150.
%C A283151 With A283150 and A248324, forms doubly infinite Riordan array. For b and c the sequences A283150 and A248324, respectively, and i,j >= 0, the doubly infinite array with d(i,j) = a(i,j), d(-j,-i) = b(i,j), d(i,-j) = c(i,j), and d(-i,j) = 0 (except d(0,0)=1) is a doubly infinite Riordan array.
%H A283151 Peter Bala, <a href="/A264772/a264772_1.pdf">A 4-parameter family of embedded Riordan arrays</a>
%H A283151 Peter Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%H A283151 H. Prodinger, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-5/prodinger.pdf">Some information about the binomial transform</a>, The Fibonacci Quarterly, 32, 1994, 412-415.
%H A283151 Thomas M. Richardson, <a href="http://arxiv.org/abs/1609.01193">The three 'R's and Dual Riordan Arrays</a>, arXiv:1609.01193 [math.CO], 2016.
%F A283151 a(m,n) = binomial(-n-2/3, m-n)*(-1)^m*9^(m-n).
%F A283151 G.f.: (1-9x)^(1/3)/(xy-9x+1).
%F A283151 Recurrence: a(m,n) = a(m,n-1)*(n-1-m)/(9*n-3) for 0 < n <= m; matrix inverse of a(m,n) is a(m,n). - _Werner Schulte_, Aug 05 2017
%F A283151 From _Peter Bala_, Mar 05 2018 (Start):
%F A283151 Let P(n,x) = Sum_{k = 0..n} T(n,k)*x^(n-k) denote the n-th row polynomial in descending powers of x. Then (-1)^n*P(n,x) is the n-th degree Taylor polynomial of (1 - 9*x)^(n-1/3) about 0. For example, for n = 4 we have (1 - 9*x)^(11/3) = 2970*x^4 - 1980*x^3 + 396*x^2 - 33*x + 1 + O(x^5).
%F A283151 Let R(n,x) denote the n-th row polynomial of this triangle. The polynomial R(n,9*x) has the e.g.f. Sum_{k = 0..n} T(n,k)*(9*x)^k/k!. The e.g.f. for the n-th diagonal of the triangle (starting at n = 0 for the main diagonal) equals exp(-x) * the e.g.f. for the polynomial R(n,9*x). For example, when n = 3 we have exp(-x)*(360 - 180*(9*x) + 24*(9*x)^2/2! - (9*x)^3/3!) = 360 - 1980*x + 5544*x^2/2! - 11781*x^3/3! + 21420*x^4/4! - ....
%F A283151 Let F(x) = (1 - ( 1 - 9*x)^(1/3))/(3*x). See A025748. The derivatives of F(x) are related to the row polynomials P(n,x) by the identity x^n/n! * (d/dx)^n(F(x)) = 1/(3*x)*( (-1)^n - P(n,x)/(1 - 9*x)^(n-1/3) ), n = 0,1,2,.... Cf. A283151 and A046521. (End)
%F A283151 From _Peter Bala_, Aug 18 2021: (Start)
%F A283151 T(n,k) = (-1)^k*binomial(n-1/3, n-k)*9^(n-k).
%F A283151 Analogous to the binomial transform we have the following sequence transformation formula: g(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*f(k) iff f(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*g(k). See Prodinger, bottom of p. 413, with b replaced with 9*b, c = -1 and d = 2/3.
%F A283151 Equivalently, if F(x) = Sum_{n >= 0} f(n)*x^n and G(x) = Sum_{n >= 0} g(n)*x^n are a pair of formal power series then
%F A283151 G(x) = (1/(1 - 9*b*x)^(2/3)) * F(x/(1 - 9*b*x)) iff F(x) = (1/(1 + 9*b*x)^(2/3)) * G(x/(1 + 9*b*x)).
%F A283151 The infinitesimal generator of the unsigned array has the sequence (9*n+6) n>=0 on the main subdiagonal and zeros elsewhere. The m-th power of the unsigned array has entries m^(n-k)*|T(n,k)|. (End)
%e A283151 Triangle begins
%e A283151          1;
%e A283151          6,        -1;
%e A283151         45,       -15,       1;
%e A283151        360,      -180,      24,       -1;
%e A283151       2970,     -1980,     396,      -33,      1;
%e A283151      24948,    -20790,    5544,     -693,     42,     -1;
%e A283151     212058,   -212058,   70686,   -11781,   1071,    -51,    1;
%e A283151    1817640,  -2120580,  848232,  -176715,  21420,  -1530,   60,  -1;
%e A283151   15677145, -20902860, 9754668, -2438667, 369495, -35190, 2070, -69, 1;
%Y A283151 Cf. A004988, A248324, A283150, A025748, A046521.
%K A283151 sign,tabl,easy
%O A283151 0,2
%A A283151 _Tom Richardson_, Mar 01 2017
%E A283151 Offset corrected by _Werner Schulte_, Aug 05 2017