Real-rooted atlas

A123519 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-18 - 15\,x + 6\,x^{2} - 12\,n - 10\,n x + 13\,n x^{2} + 9\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-42 - 95\,x - 26\,x^{2} + 12\,x^{3} - 24\,n - 62\,n x + 9\,n x^{2} + 22\,n x^{3} - 6\,n^{2} x + 19\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-18 - 54\,x - 28\,x^{2} + 14\,x^{3} + 6\,x^{4} + 24\,n x^{2} + 32\,n x^{3} + 8\,n x^{4} + 6\,n^{2} x^{2} + 8\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 11 + 28\,x + 12\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
011281251
14115213232357
2153744972480802429
3571343659884480152019216187
42131152843328833408172004416448106175
579536622417301621792015080058752120961024687785
629681281424857448129964811308005925121850243174423044410585
711077111781964101048726720076215205025216211052854886480640512028049103

All 15 rows

Technical details
Node id
n-8f29ab3b1545712d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-35714166ab2fcc1d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      11,       28,       12;
  1:      41,      152,      132,        32;
  2:     153,      744,      972,       480,        80;
  3:     571,     3436,     5988,      4480,      1520,       192;
  4:    2131,    15284,    33288,     33408,     17200,      4416,       448;
  5:    7953,    66224,   173016,    217920,    150800,     58752,     12096,     1024;
  6:   29681,   281424,   857448,   1299648,   1130800,    592512,    185024,    31744,     2304;
  7:  110771,  1178196,  4101048,   7267200,   7621520,   5025216,   2110528,   548864,    80640,    5120;
  8:  413403,  4874444, 19081332,  38705280,  47523280,  37800000,  19982592,  6987776,  1555200,  199680,  11264;
  9: 1542841, 19973192, 86847612, 198437472, 279234800, 260230656, 166019840, 73076736, 21862656, 4249600, 484352, 24576;
%F Formula
The row polynomials satisfy -(12*n+18+(10*n+15)*x-(2*n^3+9*n^2+13*n+6)*x^2)*P(n,x) = -(24*n+42+(6*n^2+62*n+95)*x-(4*n^3+19*n^2+9*n-26)*x^2-(2*n^3+12*n^2+22*n+12)*x^3)*P(n-1,x) - (18+54*x-(6*n^2+24*n-28)*x^2-(8*n^2+32*n+14)*x^3-(2*n^2+8*n+6)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 28*x + 12*x^2.
%t Mathematica
P[0, x_] := 11 + 28*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n + 42 + (6*n^2 + 62*n + 95)*x - (4*n^3 + 19*n^2 + 9*n - 26)*x^2 - (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*P[n - 1, x] - (18 + 54*x - (6*n^2 + 24*n - 28)*x^2 - (8*n^2 + 32*n + 14)*x^3 - (2*n^2 + 8*n + 6)*x^4)*D[P[n - 1, x], x])/(-12*n - 18 - (10*n + 15)*x + (2*n^3 + 9*n^2 + 13*n + 6)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 28*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(24*n + 42 + (6*n^2 + 62*n + 95)*x - (4*n^3 + 19*n^2 + 9*n - 26)*x^2 - (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*v[i - 1] - (18 + 54*x - (6*n^2 + 24*n - 28)*x^2 - (8*n^2 + 32*n + 14)*x^3 - (2*n^2 + 8*n + 6)*x^4)*deriv(v[i - 1], x))/(-12*n - 18 - (10*n + 15)*x + (2*n^3 + 9*n^2 + 13*n + 6)*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