Real-rooted atlas

A257609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(8 - 36\,x + 56\,x^{2} - 36\,x^{3} + 8\,x^{4} - 8\,n + 28\,n x - 48\,n x^{2} + 40\,n x^{3} - 8\,n x^{4} - 8\,n^{2} x + 20\,n^{2} x^{2} - 12\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 20\,x^{2} + 36\,x^{3} - 28\,x^{4} + 8\,x^{5} - 4\,n x + 16\,n x^{2} - 24\,n x^{3} + 12\,n x^{4} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-10 + 24\,x - 18\,x^{2} + 4\,x^{3} + 6\,n - 14\,n x + 10\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 8\,x^{2} - 4\,x^{3} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22024448
3962641128480
443225282304480165760
5185621152386561696019843280640
677441601285400964595201146248064641290240
73174411306246547200999616047467527392003251212823224320
8128768760780871503360182526464152924160450257924639232130560256464486400
9519168495068167236689922916663296402540748820692510724045168642864332852326451210218700800
102085888314685440692541542442143686656904799784967593260236825766430720350677401617506406420951041024245248819200
118364032196736000063572635648564145317888179994604339223297761320961283566804992302735265792296798699521063118848838451220486376469299200
12335011841215541248056557599129671215496806403256888965120062081652867072524882844057602003711503564834123062681602471145881606429425664335462404096178541140377600
13134103040744734474244913313153024858962287493125471277587660801480051256647680183311899636531210708359495352322948214164520963732152348672020342475980803877881446413420134481925356234211328000
145366251524535456563204191186903040099963713939046486659272566538243229454482463129656429106310594560481466712843878402025685956629299241482533951569923992818409963521661317578752023348043776053683814416384171399494762496000
Technical details
Node id
n-f09291b462bcd6d0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d762c6db5469cb8f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:      4,        2;
  2:     20,       24,         4;
  3:     96,      264,       112,          8;
  4:    432,     2528,      2304,        480,         16;
  5:   1856,    21152,     38656,      16960,       1984,         32;
  6:   7744,   160128,    540096,     459520,     114624,       8064,        64;
  7:  31744,  1130624,   6547200,    9996160,    4746752,     739200,     32512,      128;
  8: 128768,  7607808,  71503360,  182526464,  152924160,   45025792,   4639232,   130560,    256;
  9: 519168, 49506816, 723668992, 2916663296, 4025407488, 2069251072, 404516864, 28643328, 523264, 512;
%F Formula
The row polynomials satisfy (n-2+2*x)*P(n,x) = -(8*n-8+(8*n^2-28*n+36)*x+(4*n^3-20*n^2+48*n-56)*x^2+(12*n^2-40*n+36)*x^3+(8*n-8)*x^4)*P(n-2,x) - ((4*n-4)*x+(4*n^2-16*n+20)*x^2-(4*n^2-24*n+36)*x^3-(12*n-28)*x^4-8*x^5)*d/dx P(n-2,x) + (6*n-10+(4*n^2-14*n+24)*x+(10*n-18)*x^2+4*x^3)*P(n-1,x) + ((2*n-4)*x-(2*n-8)*x^2-4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 8 + (8*n^2 - 28*n + 36)*x + (4*n^3 - 20*n^2 + 48*n - 56)*x^2 + (12*n^2 - 40*n + 36)*x^3 + (8*n - 8)*x^4)*P[n - 2, x] - ((4*n - 4)*x + (4*n^2 - 16*n + 20)*x^2 - (4*n^2 - 24*n + 36)*x^3 - (12*n - 28)*x^4 - 8*x^5)*D[P[n - 2, x], x] + (6*n - 10 + (4*n^2 - 14*n + 24)*x + (10*n - 18)*x^2 + 4*x^3)*P[n - 1, x] + ((2*n - 4)*x - (2*n - 8)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*x)]];
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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n - 8 + (8*n^2 - 28*n + 36)*x + (4*n^3 - 20*n^2 + 48*n - 56)*x^2 + (12*n^2 - 40*n + 36)*x^3 + (8*n - 8)*x^4)*v[i - 2] - ((4*n - 4)*x + (4*n^2 - 16*n + 20)*x^2 - (4*n^2 - 24*n + 36)*x^3 - (12*n - 28)*x^4 - 8*x^5)*deriv(v[i - 2], x) + (6*n - 10 + (4*n^2 - 14*n + 24)*x + (10*n - 18)*x^2 + 4*x^3)*v[i - 1] + ((2*n - 4)*x - (2*n - 8)*x^2 - 4*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*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

Same fingerprint ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.