Real-rooted atlas

A113953 → basis A013611 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(4 + 16\,x - 12\,n - 16\,n x + 8\,n^{2}\right)\,P_{n}(x) = \left(-160 - 288\,x - 128\,x^{2} + 168\,n + 160\,n x - 32\,n^{2}\right)\,P_{n-1}(x) + \left(-27 - 156\,x - 208\,x^{2} - 64\,x^{3} + 18\,n + 80\,n x + 32\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2163248
344224192460
4120896192010243960
529230081036813312512032100
6704905643008911368192024576250400
71628253761541764577286809604669441146881901500
8372067776501120192102440448004571136252313652428814157000
9834017433615206407153664195891203145728028442624131072002359296103812500
1018512436128438220824451072826572801734574082229534721672478726606028810485760752150000
11406521066336121322887835033631562752082118246413829079041472724992941359104325058560461373445396587500
12886002560000325282562387015681117204480347661926472799354881016751718492059729925117050880156866969620132659238408175000
1319174860519048494886469829734437254707201351158988833934344192590364016647009704345655050240000270364835847449083904872415232271502562500
1441264014123488217036800197572915211838382080490722508801439967477763010592768004456804515844586576281603174249267201395193282563489660928037580963841908111000000
Technical details
Node id
n-46e5d5dc7ec30701 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5ca9878f2e88af9a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    0;
  1:    4;
  2:   16,     32;
  3:   44,    224,     192;
  4:  120,    896,    1920,    1024;
  5:  292,   3008,   10368,   13312,     5120;
  6:  704,   9056,   43008,   91136,    81920,    24576;
  7: 1628,  25376,  154176,  457728,   680960,   466944,   114688;
  8: 3720,  67776,  501120, 1921024,  4044800,  4571136,  2523136,   524288;
  9: 8340, 174336, 1520640, 7153664, 19589120, 31457280, 28442624, 13107200, 2359296;
%F Formula
The row polynomials satisfy (8*n^2-12*n+4-(16*n-16)*x)*P(n,x) = -(32*n^2-168*n+160-(160*n-288)*x+128*x^2)*P(n-1,x) + (18*n-27+(80*n-156)*x+(32*n-208)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(32*n^2 - 168*n + 160 - (160*n - 288)*x + 128*x^2)*P[n - 1, x] + (18*n - 27 + (80*n - 156)*x + (32*n - 208)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(8*n^2 - 12*n + 4 - (16*n - 16)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(32*n^2 - 168*n + 160 - (160*n - 288)*x + 128*x^2)*v[i - 1] + (18*n - 27 + (80*n - 156)*x + (32*n - 208)*x^2 - 64*x^3)*deriv(v[i - 1], x))/(8*n^2 - 12*n + 4 - (16*n - 16)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.