Real-rooted atlas

A091044 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - n x + 4\,n^{2} - 2\,n^{2} x + 4\,n^{3} + 6\,n^{3} x + n^{3} x^{2} + 12\,n^{4} x + 4\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-8\,x - 2\,x^{2} - 2\,x^{3} + 4\,n - 12\,n x - 44\,n x^{2} - 10\,n x^{3} + 16\,n^{2} + 56\,n^{2} x - 2\,n^{2} x^{2} - 10\,n^{2} x^{3} + 16\,n^{3} + 120\,n^{3} x + 172\,n^{3} x^{2} + 62\,n^{3} x^{3} + 48\,n^{4} x + 136\,n^{4} x^{2} + 80\,n^{4} x^{3} + 16\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} - 6\,x^{3} - 4\,x^{4} - 8\,n x - 40\,n x^{2} - 62\,n x^{3} - 28\,n x^{4} - 8\,n^{2} x - 64\,n^{2} x^{2} - 134\,n^{2} x^{3} - 76\,n^{2} x^{4} - 24\,n^{3} x^{2} - 80\,n^{3} x^{3} - 64\,n^{3} x^{4} - 8\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141620
216128108252
36476814405122784
425640961209610240200028688
510242048082944114688560006912282048
6409698304506880983040806400258048219522678720
716384458752287539272089608448000464486410536966553624771584

All 15 rows

Technical details
Node id
n-81d6758fbe6438c0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-09c73d06b6628b2b

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,      16;
  2:     16,     128,      108;
  3:     64,     768,     1440,       512;
  4:    256,    4096,    12096,     10240,      2000;
  5:   1024,   20480,    82944,    114688,     56000,      6912;
  6:   4096,   98304,   506880,    983040,    806400,    258048,     21952;
  7:  16384,  458752,  2875392,   7208960,   8448000,   4644864,   1053696,     65536;
  8:  65536, 2097152, 15482880,  47710208,  73216000,  58392576,  23181312,   3932160,   186624;
  9: 262144, 9437184, 80216064, 293601280, 559104000, 590413824, 344408064, 103809024, 13685760, 512000;
%F Formula
The row polynomials satisfy (4*n^3+4*n^2+n+(12*n^4+6*n^3-2*n^2-n)*x+(4*n^5+4*n^4+n^3)*x^2)*P(n,x) = (16*n^3+16*n^2+4*n+(48*n^4+120*n^3+56*n^2-12*n-8)*x+(16*n^5+136*n^4+172*n^3-2*n^2-44*n-2)*x^2+(24*n^5+80*n^4+62*n^3-10*n^2-10*n-2)*x^3)*P(n-1,x) - ((8*n^2+8*n+2)*x+(24*n^3+64*n^2+40*n+6)*x^2+(8*n^4+80*n^3+134*n^2+62*n+6)*x^3+(16*n^4+64*n^3+76*n^2+28*n+4)*x^4)*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[((16*n^3 + 16*n^2 + 4*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (24*n^5 + 80*n^4 + 62*n^3 - 10*n^2 - 10*n - 2)*x^3)*P[n - 1, x] - ((8*n^2 + 8*n + 2)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (16*n^4 + 64*n^3 + 76*n^2 + 28*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 4*n^2 + n + (12*n^4 + 6*n^3 - 2*n^2 - n)*x + (4*n^5 + 4*n^4 + n^3)*x^2)]];
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] = ((16*n^3 + 16*n^2 + 4*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (24*n^5 + 80*n^4 + 62*n^3 - 10*n^2 - 10*n - 2)*x^3)*v[i - 1] - ((8*n^2 + 8*n + 2)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (16*n^4 + 64*n^3 + 76*n^2 + 28*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 4*n^2 + n + (12*n^4 + 6*n^3 - 2*n^2 - n)*x + (4*n^5 + 4*n^4 + n^3)*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