Real-rooted atlas

A122076 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-200 + 130\,x - 82\,x^{2} - 4\,x^{3} + 200\,n - 630\,n x + 562\,n x^{2} + 28\,n x^{3} + 500\,n^{2} x - 1080\,n^{2} x^{2} + 160\,n^{2} x^{3} - 8\,n^{2} x^{4} + 600\,n^{3} x^{2} - 424\,n^{3} x^{3} + 40\,n^{3} x^{4} + 240\,n^{4} x^{3} - 64\,n^{4} x^{4} + 32\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-800 - 830\,x + 782\,x^{2} + 1160\,x^{3} + 300\,x^{4} - 24\,x^{5} + 800\,n - 1870\,n x - 3662\,n x^{2} - 1244\,n x^{3} + 176\,n x^{4} + 88\,n x^{5} + 2700\,n^{2} x - 120\,n^{2} x^{2} - 3140\,n^{2} x^{3} - 1100\,n^{2} x^{4} - 24\,n^{2} x^{5} + 3000\,n^{3} x^{2} + 2104\,n^{3} x^{3} - 448\,n^{3} x^{4} - 200\,n^{3} x^{5} + 1120\,n^{4} x^{3} + 944\,n^{4} x^{4} + 64\,n^{4} x^{5} + 128\,n^{5} x^{4} + 96\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-125 - 75\,x + 565\,x^{2} + 961\,x^{3} + 530\,x^{4} + 68\,x^{5} - 16\,x^{6} - 750\,n x - 1550\,n x^{2} - 760\,n x^{3} + 260\,n x^{4} + 268\,n x^{5} + 48\,n x^{6} - 1200\,n^{2} x^{2} - 2600\,n^{2} x^{3} - 1732\,n^{2} x^{4} - 316\,n^{2} x^{5} + 16\,n^{2} x^{6} - 600\,n^{3} x^{3} - 1240\,n^{3} x^{4} - 768\,n^{3} x^{5} - 128\,n^{3} x^{6} - 80\,n^{4} x^{4} - 144\,n^{4} x^{5} - 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 3 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
13811
27321857
31812013532305
447416720384501617
5123136031952720875728345
632242721269014912787517289841897
784313048468727033651975191523087128205441

All 15 rows

Technical details
Node id
n-d3d7c00ff700f502 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ab269ce9163b66d2

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:    2;
  1:    3,      8;
  2:    7,     32,     18;
  3:   18,    120,    135,      32;
  4:   47,    416,    720,     384,      50;
  5:  123,   1360,   3195,    2720,     875,     72;
  6:  322,   4272,  12690,   14912,    7875,   1728,     98;
  7:  843,  13048,  46872,   70336,   51975,  19152,   3087,   128;
  8: 2207,  39040, 164520,  300544,  286000, 149184,  41160,  5120,  162;
  9: 5778, 114984, 556065, 1197408, 1395225, 944136, 372498, 80640, 8019, 200;
%F Formula
The row polynomials satisfy (200*n-200+(500*n^2-630*n+130)*x+(600*n^3-1080*n^2+562*n-82)*x^2+(240*n^4-424*n^3+160*n^2+28*n-4)*x^3+(32*n^5-64*n^4+40*n^3-8*n^2)*x^4)*P(n,x) = (800*n-800+(2700*n^2-1870*n-830)*x+(3000*n^3-120*n^2-3662*n+782)*x^2+(1120*n^4+2104*n^3-3140*n^2-1244*n+1160)*x^3+(128*n^5+944*n^4-448*n^3-1100*n^2+176*n+300)*x^4+(96*n^5+64*n^4-200*n^3-24*n^2+88*n-24)*x^5)*P(n-1,x) - (125+(750*n+75)*x+(1200*n^2+1550*n-565)*x^2+(600*n^3+2600*n^2+760*n-961)*x^3+(80*n^4+1240*n^3+1732*n^2-260*n-530)*x^4+(144*n^4+768*n^3+316*n^2-268*n-68)*x^5+(64*n^4+128*n^3-16*n^2-48*n+16)*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((800*n - 800 + (2700*n^2 - 1870*n - 830)*x + (3000*n^3 - 120*n^2 - 3662*n + 782)*x^2 + (1120*n^4 + 2104*n^3 - 3140*n^2 - 1244*n + 1160)*x^3 + (128*n^5 + 944*n^4 - 448*n^3 - 1100*n^2 + 176*n + 300)*x^4 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^5)*P[n - 1, x] - (125 + (750*n + 75)*x + (1200*n^2 + 1550*n - 565)*x^2 + (600*n^3 + 2600*n^2 + 760*n - 961)*x^3 + (80*n^4 + 1240*n^3 + 1732*n^2 - 260*n - 530)*x^4 + (144*n^4 + 768*n^3 + 316*n^2 - 268*n - 68)*x^5 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^6)*D[P[n - 1, x], x])/(200*n - 200 + (500*n^2 - 630*n + 130)*x + (600*n^3 - 1080*n^2 + 562*n - 82)*x^2 + (240*n^4 - 424*n^3 + 160*n^2 + 28*n - 4)*x^3 + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((800*n - 800 + (2700*n^2 - 1870*n - 830)*x + (3000*n^3 - 120*n^2 - 3662*n + 782)*x^2 + (1120*n^4 + 2104*n^3 - 3140*n^2 - 1244*n + 1160)*x^3 + (128*n^5 + 944*n^4 - 448*n^3 - 1100*n^2 + 176*n + 300)*x^4 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^5)*v[i - 1] - (125 + (750*n + 75)*x + (1200*n^2 + 1550*n - 565)*x^2 + (600*n^3 + 2600*n^2 + 760*n - 961)*x^3 + (80*n^4 + 1240*n^3 + 1732*n^2 - 260*n - 530)*x^4 + (144*n^4 + 768*n^3 + 316*n^2 - 268*n - 68)*x^5 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^6)*deriv(v[i - 1], x))/(200*n - 200 + (500*n^2 - 630*n + 130)*x + (600*n^3 - 1080*n^2 + 562*n - 82)*x^2 + (240*n^4 - 424*n^3 + 160*n^2 + 28*n - 4)*x^3 + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*x^4)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.