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A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

%I A207538 #48 Mar 07 2023 11:20:05
%S A207538 1,2,4,1,8,4,16,12,1,32,32,6,64,80,24,1,128,192,80,8,256,448,240,40,1,
%T A207538 512,1024,672,160,10,1024,2304,1792,560,60,1,2048,5120,4608,1792,280,
%U A207538 12,4096,11264,11520,5376,1120,84,1,8192,24576,28160,15360
%N A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.
%C A207538 As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is the triangle given by (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 04 2012
%C A207538 The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and  along (first layer) skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. - _Zagros Lalo_, Jul 31 2018
%C A207538 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.414213562373095... (A014176: Decimal expansion of the silver mean, 1+sqrt(2)), when n approaches infinity. - _Zagros Lalo_, Jul 31 2018
%D A207538 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358.
%H A207538 Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H A207538 S. Halici, <a href="http://www.emis.de/journals/AUA/acta29/Paper9-Acta29-2012.pdf">On some Pell polynomials </a>, Acta Universitatis Apulensis, No. 29/2012, pp. 105-112.
%H A207538 Zagros Lalo, <a href="/A207538/a207538.pdf">First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n</a>
%H A207538 Zagros Lalo, <a href="/A207538/a207538_1.pdf">First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n</a>
%F A207538 u(n,x) = u(n-1,x)+(x+1)*v(n-1,x), v(n,x) = u(n-1,x)+v(n-1,x), where u(1,x) = 1, v(1,x) = 1. Also, A207538 = |A133156|.
%F A207538 From _Philippe Deléham_, Mar 04 2012: (Start)
%F A207538 With 0<=k<=n:
%F A207538 Mirror image of triangle in A099089.
%F A207538 Skew version of A038207.
%F A207538 Riordan array (1/(1-2*x), x^2/(1-2*x)).
%F A207538 G.f.: 1/(1-2*x-y*x^2).
%F A207538 Sum_{k, 0<=k<=n} T(n,k)*x^k = A190958(n+1), A127357(n), A090591(n), A089181(n+1), A088139(n+1), A045873(n+1), A088138(n+1), A088137(n+1), A099087(n), A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively.
%F A207538 T(n,k) = 2*T(n-1,k) + T(-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n, k) = 0 if k<0 or if k>n. (End)
%F A207538 T(n,k) = A013609(n-k, n-2*k+1). - _Johannes W. Meijer_, Sep 05 2013
%F A207538 From _Tom Copeland_, Feb 11 2016: (Start)
%F A207538 A053117 is a reflected, aerated and signed version of this entry. This entry belongs to a family discussed in A097610 with parameters h1 = -2 and h2 = -y.
%F A207538 Shifted o.g.f.: G(x,t) = x / (1 - 2 x - t x^2).
%F A207538 The compositional inverse of G(x,t) is Ginv(x,t) = -[(1 + 2x) - sqrt[(1+2x)^2 + 4t x^2]] / (2tx) = x - 2 x^2 + (4-t) x^3 - (8-6t) x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0).
%F A207538 (End)
%e A207538 First seven rows:
%e A207538 1
%e A207538 2
%e A207538 4...1
%e A207538 8...4
%e A207538 16..12..1
%e A207538 32..32..6
%e A207538 64..80..24..1
%e A207538 (2, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, ...) begins:
%e A207538     1
%e A207538     2,   0
%e A207538     4,   1,  0
%e A207538     8,   4,  0, 0
%e A207538    16,  12,  1, 0, 0
%e A207538    32,  32,  6, 0, 0, 0
%e A207538    64,  80, 24, 1, 0, 0, 0
%e A207538   128, 192, 80, 8, 0, 0, 0, 0
%t A207538 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207538 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
%t A207538 v[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207538 Table[Factor[u[n, x]], {n, 1, z}]
%t A207538 Table[Factor[v[n, x]], {n, 1, z}]
%t A207538 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207538 TableForm[cu]
%t A207538 Flatten[%]  (* A207537, |A028297| *)
%t A207538 Table[Expand[v[n, x]], {n, 1, z}]
%t A207538 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207538 TableForm[cv]
%t A207538 Flatten[%]  (* A207538, |A133156| *)
%t A207538 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* _Zagros Lalo_, Jul 31 2018 *)
%t A207538 t[n_, k_] := t[n, k] = 2^(n - 2 k) * (n -  k)!/((n - 2 k)! k!) ; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]} ]  // Flatten (* _Zagros Lalo_, Jul 31 2018 *)
%Y A207538 Cf. A028297, A207537, A133156, A038207, A099089.
%Y A207538 Cf. A053117, A097610, A091894.
%Y A207538 Cf. A013609, A038207.
%Y A207538 Cf. A128099.
%K A207538 nonn,tabf
%O A207538 1,2
%A A207538 _Clark Kimberling_, Feb 18 2012