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 A283150 #44 Jan 05 2025 19:51:41
%S A283150 1,3,-1,18,-12,1,126,-126,21,-1,945,-1260,315,-30,1,7371,-12285,4095,
%T A283150 -585,39,-1,58968,-117936,49140,-9360,936,-48,1,480168,-1120392,
%U A283150 560196,-133380,17784,-1368,57,-1,3961386,-10563696,6162156,-1760616,293436,-30096,1881,-66,1,33011550,-99034650,66023100
%N A283150 Riordan array (1/(1-9x)^(1/3), x/(9x-1)).
%C A283150 Triangle read by rows. This is an example of a Riordan group involution. Dual Riordan array of A283151. With A283151 and A248324, forms doubly infinite Riordan array. For b and c the sequences A283151 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(j,i), and d(-i,j) = 0 (except d(0,0)=1) is a doubly infinite Riordan array.
%C A283150 Matrix inverse of a(m,n) is a(m,n). - _Werner Schulte_, Aug 05 2017
%H A283150 Peter Bala, <a href="/A264772/a264772_1.pdf">A 4-parameter family of embedded Riordan arrays</a>
%H A283150 Peter Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%H A283150 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 A283150 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 A283150 a(m,n) = binomial(-n-1/3, m-n)*(-1)^m*9^(m-n).
%F A283150 G.f.: (1-9x)^(2/3)/(xy-9x+1).
%F A283150 Recurrence: a(m,n) = a(m, n-1)*(n-1-m)/(9*n-6) for 0 < n <= m. - _Werner Schulte_, Aug 05 2017
%F A283150 From _Peter Bala_, Mar 05 2018 (Start):
%F A283150 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-2/3) about 0. For example, for n = 4 we have (1 - 9*x)^(10/3) = 945*x^4 - 1260*x^3 + 315*x^2 - 30*x + 1 + O(x^5).
%F A283150 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)*(126 - 126*(9*x) + 21*(9*x)^2/2! - (9*x)^3/3!) = 126 - 1260*x + 4095*x^2/2! - 9360*x^3/3! + 17784*x^4/4! - ....
%F A283150 Let F(x) = (1 - ( 1 - 9*x)^(2/3))/(3*x) denote the o.g.f. of A155579. 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-2/3) ), n = 0,1,2,.... Cf. A283151 and A046521. (End)
%F A283150 From _Peter Bala_, Aug 18 2021: (Start)
%F A283150 T(n,k) = (-1)^k*binomial(n-2/3, n-k)*9^(n-k).
%F A283150 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 = 1/3.
%F A283150 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 A283150 G(x) = (1/(1 - 9*b*x)^(1/3)) * F(x/(1 - 9*b*x)) iff F(x) = (1/(1 + 9*b*x)^(1/3)) * G(x/(1 + 9*b*x)).
%F A283150 The infinitesimal generator of the unsigned array has the sequence (9*n+3) 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 A283150 The triangle begins
%e A283150 1;
%e A283150 3, -1;
%e A283150 18, -12, 1;
%e A283150 126, -126, 21, -1;
%e A283150 945, -1260, 315, -30, 1;
%e A283150 7371, -12285, 4095, -585, 39, -1;
%e A283150 58968, -117936, 49140, -9360, 936, -48, 1;
%e A283150 480168, -1120392, 560196, -133380, 17784, -1368, 57, -1;
%e A283150 3961386, -10563696, 6162156, -1760616, 293436, -30096, 1881, -66, 1;
%p A283150 T := (n, k) -> (-1)^k*binomial(n - 2/3, n - k)*9^(n - k):
%p A283150 for n from 0 to 6 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Sep 03 2021
%o A283150 (PARI) a(m,n) = binomial(-n-1/3, m-n)*(-1)^m*9^(m-n);
%o A283150 tabl(nn) = for(n=0, nn, for (k=0, n, print1(a(n, k), ", ")); print); \\ _Michel Marcus_, Aug 07 2017
%Y A283150 Cf. A004987, A248324, A283151, A046521, A155579.
%K A283150 sign,tabl,easy
%O A283150 0,2
%A A283150 _Tom Richardson_, Mar 01 2017
%E A283150 Offset corrected by _Werner Schulte_, Aug 05 2017