Real-rooted atlas

A110441 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 2\,n x + n^{3} x^{2}\right)\,P_{n}(x) = \left(-8 - 10\,x - 4\,x^{2} - 8\,n - 12\,n x - 4\,n x^{2} + n x^{3} - 2\,n^{2} x + 2\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4 - 12\,x - 10\,x^{2} - 2\,x^{3} + 2\,n x^{2} + 4\,n x^{3} + n x^{4} + 2\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
214321258
34817614432400
41648321080512802668
5560363265284864160019217376
61912151043484835840184004608448111160
765286080017164822656016000062208125441024701312
822288239104799296129228811696006236161944323276823044375696
9760969239683571200684748875891205177088220774457344082944512027054208
10259808352204815458112343285764513600037675008205219847274496161740820480011264166009504
11887040132794886525619216484096025118720024867225616558080074858496226713604403200495616245761012157184
1230285444962201626994700876472320013265792001522188288120151449666335539225588224067584000116469761179648532486137304256
1310340096184043264109814169634490695686715750400877519872080266690565250023424247073587283015680019428147230081024276889611468837037374976
1435303296678289408440480256015195209728328325171204817421926450179110912380200550402118373171286759833602580176896541851648761446406422528245760222584063872
Technical details
Node id
n-f24b03ba37099aab · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e8387f91b8c28d2d

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,      4;
  2:    14,     32,      12;
  3:    48,    176,     144,      32;
  4:   164,    832,    1080,     512,      80;
  5:   560,   3632,    6528,    4864,    1600,     192;
  6:  1912,  15104,   34848,   35840,   18400,    4608,     448;
  7:  6528,  60800,  171648,  226560,  160000,   62208,   12544,   1024;
  8: 22288, 239104,  799296, 1292288, 1169600,  623616,  194432,  32768,  2304;
  9: 76096, 923968, 3571200, 6847488, 7589120, 5177088, 2207744, 573440, 82944, 5120;
%F Formula
The row polynomials satisfy -(4*n+2*n*x-n^3*x^2)*P(n,x) = -(8*n+8+(2*n^2+12*n+10)*x-(2*n^3+2*n^2-4*n-4)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (4+12*x-(2*n^2+2*n-10)*x^2-(4*n^2+4*n-2)*x^3-(n^2+n)*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[(-(8*n + 8 + (2*n^2 + 12*n + 10)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (4 + 12*x - (2*n^2 + 2*n - 10)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-4*n - 2*n*x + 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] = (-(8*n + 8 + (2*n^2 + 12*n + 10)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (4 + 12*x - (2*n^2 + 2*n - 10)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-4*n - 2*n*x + 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