Real-rooted atlas

A110441 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 2\,n x + 4\,n x^{2} - 20\,n^{3} x^{2} - 20\,n^{3} x^{3} + 4\,n^{5} x^{4}\right)\,P_{n}(x) = \left(8 + 22\,x + 28\,x^{2} + 8\,x^{3} - 16\,x^{4} + 8\,n + 36\,n x + 24\,n x^{2} - 60\,n x^{3} - 88\,n x^{4} + 14\,n^{2} x - 44\,n^{2} x^{2} - 168\,n^{2} x^{3} - 144\,n^{2} x^{4} + 8\,n^{2} x^{5} - 40\,n^{3} x^{2} - 108\,n^{3} x^{3} - 72\,n^{3} x^{4} + 24\,n^{3} x^{5} - 8\,n^{4} x^{3} + 8\,n^{4} x^{4} + 24\,n^{4} x^{5} + 8\,n^{5} x^{4} + 8\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(1 + 6\,x + 10\,x^{2} - 16\,x^{4} - 8\,x^{5} - 14\,n x^{2} - 76\,n x^{3} - 108\,n x^{4} - 40\,n x^{5} - 14\,n^{2} x^{2} - 76\,n^{2} x^{3} - 104\,n^{2} x^{4} - 24\,n^{2} x^{5} + 8\,n^{2} x^{6} + 8\,n^{3} x^{4} + 32\,n^{3} x^{5} + 16\,n^{3} x^{6} + 4\,n^{4} x^{4} + 16\,n^{4} x^{5} + 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141620
214128144286
348704172810243504
416433281296016384640039236
5560145287833615564812800036864413936
6191260416418176114688014720008847362007044184824
765282432002059776724992012800000119439365619712104857640971648

All 15 rows

Technical details
Node id
n-20e0e32d0efc080a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-86519ebf5f4ed87e

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:    14,     128,      144;
  3:    48,     704,     1728,      1024;
  4:   164,    3328,    12960,     16384,      6400;
  5:   560,   14528,    78336,    155648,    128000,     36864;
  6:  1912,   60416,   418176,   1146880,   1472000,    884736,    200704;
  7:  6528,  243200,  2059776,   7249920,  12800000,  11943936,   5619712,   1048576;
  8: 22288,  956416,  9591552,  41353216,  93568000, 119734272,  87105536,  33554432,   5308416;
  9: 76096, 3695872, 42854400, 219119616, 607129600, 994000896, 989069312, 587202560, 191102976, 26214400;
%F Formula
The row polynomials satisfy (4*n+2*n*x-(20*n^3-4*n)*x^2-20*n^3*x^3+4*n^5*x^4)*P(n,x) = (8*n+8+(14*n^2+36*n+22)*x-(40*n^3+44*n^2-24*n-28)*x^2-(8*n^4+108*n^3+168*n^2+60*n-8)*x^3+(8*n^5+8*n^4-72*n^3-144*n^2-88*n-16)*x^4+(8*n^5+24*n^4+24*n^3+8*n^2)*x^5)*P(n-1,x) + (1+6*x-(14*n^2+14*n-10)*x^2-(76*n^2+76*n)*x^3+(4*n^4+8*n^3-104*n^2-108*n-16)*x^4+(16*n^4+32*n^3-24*n^2-40*n-8)*x^5+(8*n^4+16*n^3+8*n^2)*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[((8*n + 8 + (14*n^2 + 36*n + 22)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (8*n^4 + 108*n^3 + 168*n^2 + 60*n - 8)*x^3 + (8*n^5 + 8*n^4 - 72*n^3 - 144*n^2 - 88*n - 16)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*P[n - 1, x] + (1 + 6*x - (14*n^2 + 14*n - 10)*x^2 - (76*n^2 + 76*n)*x^3 + (4*n^4 + 8*n^3 - 104*n^2 - 108*n - 16)*x^4 + (16*n^4 + 32*n^3 - 24*n^2 - 40*n - 8)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*D[P[n - 1, x], x])/(4*n + 2*n*x - (20*n^3 - 4*n)*x^2 - 20*n^3*x^3 + 4*n^5*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] = ((8*n + 8 + (14*n^2 + 36*n + 22)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (8*n^4 + 108*n^3 + 168*n^2 + 60*n - 8)*x^3 + (8*n^5 + 8*n^4 - 72*n^3 - 144*n^2 - 88*n - 16)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*v[i - 1] + (1 + 6*x - (14*n^2 + 14*n - 10)*x^2 - (76*n^2 + 76*n)*x^3 + (4*n^4 + 8*n^3 - 104*n^2 - 108*n - 16)*x^4 + (16*n^4 + 32*n^3 - 24*n^2 - 40*n - 8)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*deriv(v[i - 1], x))/(4*n + 2*n*x - (20*n^3 - 4*n)*x^2 - 20*n^3*x^3 + 4*n^5*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. [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