Real-rooted atlas

A111125 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-12 + 66\,x - 12\,x^{2} + 32\,n - 254\,n x + 80\,n x^{2} - 28\,n^{2} + 356\,n^{2} x - 204\,n^{2} x^{2} + 8\,n^{3} - 216\,n^{3} x + 248\,n^{3} x^{2} + 48\,n^{4} x - 144\,n^{4} x^{2} + 32\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-12 - 78\,x - 132\,x^{2} - 48\,x^{3} + 32\,n + 130\,n x - 8\,n x^{2} - 40\,n x^{3} - 28\,n^{2} + 20\,n^{2} x + 544\,n^{2} x^{2} + 252\,n^{2} x^{3} + 8\,n^{3} - 120\,n^{3} x - 536\,n^{3} x^{2} - 88\,n^{3} x^{3} + 48\,n^{4} x + 96\,n^{4} x^{2} - 176\,n^{4} x^{3} + 32\,n^{5} x^{2} + 96\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 27\,x^{2} - 50\,x^{3} - 24\,x^{4} + 8\,n x + 30\,n x^{2} - 16\,n x^{3} - 32\,n x^{4} - 4\,n^{2} x + 20\,n^{2} x^{2} + 172\,n^{2} x^{3} + 112\,n^{2} x^{4} - 24\,n^{3} x^{2} - 96\,n^{3} x^{3} - 16\,n^{4} x^{3} - 64\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211213
31204566
4128126112267
5136243480225985
6144396123213753963444
715258524964550327663711597
816081044001125013608686096037949

All 15 rows

Technical details
Node id
n-f6cc84113892da54 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f275e5e9925d5c1c

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1, 12;
   3: 1, 20,   45;
   4: 1, 28,  126,   112;
   5: 1, 36,  243,   480,   225;
   6: 1, 44,  396,  1232,  1375,   396;
   7: 1, 52,  585,  2496,  4550,  3276,    637;
   8: 1, 60,  810,  4400, 11250, 13608,   6860,   960;
   9: 1, 68, 1071,  7072, 23375, 40392,  34986, 13056,  1377;
  10: 1, 76, 1368, 10640, 43225, 97812, 122892, 80256, 23085, 1900;
%F Formula
The row polynomials satisfy (8*n^3-28*n^2+32*n-12+(48*n^4-216*n^3+356*n^2-254*n+66)*x+(32*n^5-144*n^4+248*n^3-204*n^2+80*n-12)*x^2)*P(n,x) = (8*n^3-28*n^2+32*n-12+(48*n^4-120*n^3+20*n^2+130*n-78)*x+(32*n^5+96*n^4-536*n^3+544*n^2-8*n-132)*x^2+(96*n^5-176*n^4-88*n^3+252*n^2-40*n-48)*x^3)*P(n-1,x) - ((4*n^2-8*n+4)*x+(24*n^3-20*n^2-30*n+27)*x^2+(16*n^4+96*n^3-172*n^2+16*n+50)*x^3+(64*n^4-112*n^2+32*n+24)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 120*n^3 + 20*n^2 + 130*n - 78)*x + (32*n^5 + 96*n^4 - 536*n^3 + 544*n^2 - 8*n - 132)*x^2 + (96*n^5 - 176*n^4 - 88*n^3 + 252*n^2 - 40*n - 48)*x^3)*P[n - 1, x] - ((4*n^2 - 8*n + 4)*x + (24*n^3 - 20*n^2 - 30*n + 27)*x^2 + (16*n^4 + 96*n^3 - 172*n^2 + 16*n + 50)*x^3 + (64*n^4 - 112*n^2 + 32*n + 24)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 216*n^3 + 356*n^2 - 254*n + 66)*x + (32*n^5 - 144*n^4 + 248*n^3 - 204*n^2 + 80*n - 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 120*n^3 + 20*n^2 + 130*n - 78)*x + (32*n^5 + 96*n^4 - 536*n^3 + 544*n^2 - 8*n - 132)*x^2 + (96*n^5 - 176*n^4 - 88*n^3 + 252*n^2 - 40*n - 48)*x^3)*v[i - 1] - ((4*n^2 - 8*n + 4)*x + (24*n^3 - 20*n^2 - 30*n + 27)*x^2 + (16*n^4 + 96*n^3 - 172*n^2 + 16*n + 50)*x^3 + (64*n^4 - 112*n^2 + 32*n + 24)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 216*n^3 + 356*n^2 - 254*n + 66)*x + (32*n^5 - 144*n^4 + 248*n^3 - 204*n^2 + 80*n - 12)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023