Real-rooted atlas

A316649 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 16\,x - 8\,x^{2} + 12\,n + 24\,n x + 12\,n x^{2} - 4\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 28\,x + 16\,x^{2} - 30\,n - 70\,n x - 40\,n x^{2} + 12\,n^{2} + 28\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 10\,x + 8\,x^{2}\)
\(P_{2}(x) = 26 + 150\,x + 316\,x^{2} + 288\,x^{3} + 96\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1310821
22615031628896876
32522064692412184118726080128040656
4256827240124872323256517136523840328320116480179201981632
52692835068820358726940288153911522320976024112384170472967852544212889625804899353856
6287648444080031184128131739840373001120745866592108037984011424913928756428804744857601725911043784704037847045073742848
73112896555750404577375362305765248793750073619757480704366781872005159857446455286807552449084067842722693324811949527040358952140866070118456229888262472060928

All 15 rows

Technical details
Node id
n-03c7727a6c6d39ac · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-ad0a1921d2e79f69

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:         1;
  1:         3,         10,           8;
  2:        26,        150,         316,          288,            96;
  3:       252,       2064,        6924,        12184,         11872,          6080,           1280;
  4:      2568,      27240,      124872,       323256,        517136,        523840,         328320,         116480,          17920;
  5:     26928,     350688,     2035872,      6940288,      15391152,      23209760,       24112384,       17047296,        7852544,         2128896,          258048;
  6:    287648,    4440800,    31184128,    131739840,     373001120,     745866592,     1080379840,     1142491392,      875642880,       474485760,       172591104,        37847040,        3784704;
  7:   3112896,   55575040,   457737536,   2305765248,    7937500736,   19757480704,    36678187200,    51598574464,    55286807552,     44908406784,     27226933248,     11949527040,     3589521408,      660701184,       56229888;
  8:  34013312,  689376384,  6513848704,  38094173824,  154352456832,  459534900608,  1040018086016,  1825476675456,  2511768453376,   2718687971328,   2307460571136,   1519763976192,   761578954752,   280741773312,    71805566976,   11386552320,     843448320;
  9: 374416128, 8492659200, 90533993472, 602755424256, 2808892310016, 9730580923392, 25972322718720, 54620986011648, 91762344678144, 124082812639744, 135388294150144, 118930125332480, 83494291234816, 46216232910848, 19728499867648, 6267477032960, 1395532103680, 194367979520, 12745441280;
%F Formula
The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(8*n^2-24*n+16)*x+(4*n^2-12*n+8)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(28*n^2-70*n+28)*x+(16*n^2-40*n+16)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 10*x + 8*x^2, P(2,x) = 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 10*x + 8*x^2;
P[2, x_] := 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (8*n^2 - 24*n + 16)*x + (4*n^2 - 12*n + 8)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (28*n^2 - 70*n + 28)*x + (16*n^2 - 40*n + 16)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 10*x + 8*x^2; v[3] = 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (8*n^2 - 24*n + 16)*x + (4*n^2 - 12*n + 8)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (28*n^2 - 70*n + 28)*x + (16*n^2 - 40*n + 16)*x^2)*v[i - 1])/(n^2 - 2*n)); 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