Real-rooted atlas

A172431 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2\,x + 4\,x^{2} + 12\,n x - 36\,n x^{2} + 4\,n^{2} - 98\,n^{2} x + 132\,n^{2} x^{2} - 12\,n^{3} + 204\,n^{3} x - 252\,n^{3} x^{2} + 8\,n^{4} - 168\,n^{4} x + 264\,n^{4} x^{2} + 48\,n^{5} x - 144\,n^{5} x^{2} + 32\,n^{6} x^{2}\right)\,P_{n}(x) = \left(16\,x^{2} - 16\,x^{3} - 2\,n x + 80\,n x^{2} - 40\,n x^{3} - 30\,n^{2} x + 48\,n^{2} x^{2} + 84\,n^{2} x^{3} - 4\,n^{3} - 28\,n^{3} x + 8\,n^{3} x^{2} + 84\,n^{3} x^{3} + 8\,n^{4} - 24\,n^{4} x - 312\,n^{4} x^{2} - 72\,n^{4} x^{3} + 48\,n^{5} x + 128\,n^{5} x^{2} - 144\,n^{5} x^{3} + 32\,n^{6} x^{2} + 96\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + 2\,x^{3} - 8\,x^{4} + 7\,n x^{2} + 22\,n x^{3} - 24\,n x^{4} + 2\,n^{2} x^{2} + 16\,n^{2} x^{3} + 32\,n^{2} x^{4} - 4\,n^{3} x - 4\,n^{3} x^{2} + 60\,n^{3} x^{3} + 48\,n^{3} x^{4} - 24\,n^{4} x^{2} - 96\,n^{4} x^{3} - 16\,n^{5} x^{3} - 64\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
242428
3680108194
481685403201036
510288151222407504800
612440324080647000151220268
7146245940211203150018144274480086

All 15 rows

Technical details
Node id
n-83ca5953d5765abe · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8b36a23aa5145d0b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  2;
  2:  4,   24;
  3:  6,   80,   108;
  4:  8,  168,   540,   320;
  5: 10,  288,  1512,  2240,    750;
  6: 12,  440,  3240,  8064,   7000,   1512;
  7: 14,  624,  5940, 21120,  31500,  18144,   2744;
  8: 16,  840,  9828, 45760,  99000,  99792,  41160,  4608;
  9: 18, 1088, 15120, 87360, 250250, 370656, 271656, 84480, 7290;
%F Formula
The row polynomials satisfy (8*n^4-12*n^3+4*n^2+(48*n^5-168*n^4+204*n^3-98*n^2+12*n+2)*x+(32*n^6-144*n^5+264*n^4-252*n^3+132*n^2-36*n+4)*x^2)*P(n,x) = (8*n^4-4*n^3+(48*n^5-24*n^4-28*n^3-30*n^2-2*n)*x+(32*n^6+128*n^5-312*n^4+8*n^3+48*n^2+80*n+16)*x^2+(96*n^6-144*n^5-72*n^4+84*n^3+84*n^2-40*n-16)*x^3)*P(n-1,x) - (4*n^3*x+(24*n^4+4*n^3-2*n^2-7*n-1)*x^2+(16*n^5+96*n^4-60*n^3-16*n^2-22*n-2)*x^3+(64*n^5-48*n^3-32*n^2+24*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^4 - 4*n^3 + (48*n^5 - 24*n^4 - 28*n^3 - 30*n^2 - 2*n)*x + (32*n^6 + 128*n^5 - 312*n^4 + 8*n^3 + 48*n^2 + 80*n + 16)*x^2 + (96*n^6 - 144*n^5 - 72*n^4 + 84*n^3 + 84*n^2 - 40*n - 16)*x^3)*P[n - 1, x] - (4*n^3*x + (24*n^4 + 4*n^3 - 2*n^2 - 7*n - 1)*x^2 + (16*n^5 + 96*n^4 - 60*n^3 - 16*n^2 - 22*n - 2)*x^3 + (64*n^5 - 48*n^3 - 32*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(8*n^4 - 12*n^3 + 4*n^2 + (48*n^5 - 168*n^4 + 204*n^3 - 98*n^2 + 12*n + 2)*x + (32*n^6 - 144*n^5 + 264*n^4 - 252*n^3 + 132*n^2 - 36*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^4 - 4*n^3 + (48*n^5 - 24*n^4 - 28*n^3 - 30*n^2 - 2*n)*x + (32*n^6 + 128*n^5 - 312*n^4 + 8*n^3 + 48*n^2 + 80*n + 16)*x^2 + (96*n^6 - 144*n^5 - 72*n^4 + 84*n^3 + 84*n^2 - 40*n - 16)*x^3)*v[i - 1] - (4*n^3*x + (24*n^4 + 4*n^3 - 2*n^2 - 7*n - 1)*x^2 + (16*n^5 + 96*n^4 - 60*n^3 - 16*n^2 - 22*n - 2)*x^3 + (64*n^5 - 48*n^3 - 32*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(8*n^4 - 12*n^3 + 4*n^2 + (48*n^5 - 168*n^4 + 204*n^3 - 98*n^2 + 12*n + 2)*x + (32*n^6 - 144*n^5 + 264*n^4 - 252*n^3 + 132*n^2 - 36*n + 4)*x^2)); 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. [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