Real-rooted atlas

A100861 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 + 96\,x + 32\,x^{2} - 16\,x^{3} + 80\,n x + 40\,n x^{2} - 8\,n x^{3} + 56\,n^{2} x^{2} + 28\,n^{2} x^{3} + n^{2} x^{4} + 12\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(-144\,x - 352\,x^{2} - 224\,x^{3} - 16\,x^{4} - 4\,x^{5} + 64\,n + 184\,n x + 96\,n x^{2} - 96\,n x^{3} - 80\,n x^{4} - 8\,n x^{5} + 136\,n^{2} x + 304\,n^{2} x^{2} + 176\,n^{2} x^{3} + 12\,n^{2} x^{4} - n^{2} x^{5} + 80\,n^{3} x^{2} + 144\,n^{3} x^{3} + 65\,n^{3} x^{4} + 7\,n^{3} x^{5} + 16\,n^{4} x^{3} + 22\,n^{4} x^{4} + 5\,n^{4} x^{5} + n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-32 - 192\,x - 416\,x^{2} - 416\,x^{3} - 200\,x^{4} - 48\,x^{5} - 8\,x^{6} - 96\,n x - 392\,n x^{2} - 624\,n x^{3} - 480\,n x^{4} - 176\,n x^{5} - 24\,n x^{6} - 88\,n^{2} x^{2} - 280\,n^{2} x^{3} - 322\,n^{2} x^{4} - 156\,n^{2} x^{5} - 26\,n^{2} x^{6} - 24\,n^{3} x^{3} - 60\,n^{3} x^{4} - 48\,n^{3} x^{5} - 12\,n^{3} x^{6} - 2\,n^{4} x^{4} - 4\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2246
341216
410482785
526160135321
6766008102401726
72322184378016807876

All 15 rows

Technical details
Node id
n-862661f8a222bd11 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2abb3fd695a578bb

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:    1;
  2:    2,     4;
  3:    4,    12;
  4:   10,    48,    27;
  5:   26,   160,   135;
  6:   76,   600,   810,   240;
  7:  232,  2184,  3780,  1680;
  8:  764,  8512, 18900, 13440,  2625;
  9: 2620, 33408, 88452, 80640, 23625;
%F Formula
The row polynomials satisfy (64+(80*n+96)*x+(56*n^2+40*n+32)*x^2+(12*n^3+28*n^2-8*n-16)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (64*n+(136*n^2+184*n-144)*x+(80*n^3+304*n^2+96*n-352)*x^2+(16*n^4+144*n^3+176*n^2-96*n-224)*x^3+(n^5+22*n^4+65*n^3+12*n^2-80*n-16)*x^4+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) - (32+(96*n+192)*x+(88*n^2+392*n+416)*x^2+(24*n^3+280*n^2+624*n+416)*x^3+(2*n^4+60*n^3+322*n^2+480*n+200)*x^4+(4*n^4+48*n^3+156*n^2+176*n+48)*x^5+(2*n^4+12*n^3+26*n^2+24*n+8)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((64*n + (136*n^2 + 184*n - 144)*x + (80*n^3 + 304*n^2 + 96*n - 352)*x^2 + (16*n^4 + 144*n^3 + 176*n^2 - 96*n - 224)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] - (32 + (96*n + 192)*x + (88*n^2 + 392*n + 416)*x^2 + (24*n^3 + 280*n^2 + 624*n + 416)*x^3 + (2*n^4 + 60*n^3 + 322*n^2 + 480*n + 200)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(64 + (80*n + 96)*x + (56*n^2 + 40*n + 32)*x^2 + (12*n^3 + 28*n^2 - 8*n - 16)*x^3 + (n^4 + 2*n^3 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((64*n + (136*n^2 + 184*n - 144)*x + (80*n^3 + 304*n^2 + 96*n - 352)*x^2 + (16*n^4 + 144*n^3 + 176*n^2 - 96*n - 224)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] - (32 + (96*n + 192)*x + (88*n^2 + 392*n + 416)*x^2 + (24*n^3 + 280*n^2 + 624*n + 416)*x^3 + (2*n^4 + 60*n^3 + 322*n^2 + 480*n + 200)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(64 + (80*n + 96)*x + (56*n^2 + 40*n + 32)*x^2 + (12*n^3 + 28*n^2 - 8*n - 16)*x^3 + (n^4 + 2*n^3 + 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.