Real-rooted atlas

A132812 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 2\,n^{2} + 2\,n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(4 + 12\,n + 14\,n x + 2\,n x^{2} + 8\,n^{2} + 18\,n^{2} x + 7\,n^{2} x^{2} + 4\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 6\,x^{2} - 2\,x^{3} - 4\,n x - 9\,n x^{2} - 5\,n x^{3} - 2\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
21530954
35616810816348
4210840840280252195
57923960540028806003613668
63003180183118523100787511344984364
711440800801681681601607700018480196064517352
8437583500648648641009008630630216216388083168813156597
91679601511640429624059404804586400206388053508074880486010019181520
10646646646646020785050332560803063060017153136588588012012001351357150121116167458
11249614427457584984891601790712001918620001289312645541782415079680249480023100010164144701540964
1296577001158924004589339049348653601143177750897914160465585120159629184354439804862000377520140401694226353287
13374421604867480802109241680475931456065440575205889651768358500542414898723844190266087759752089849765940481892819625407555852
14145422675203591745095828528252372896890036252591375368103235502576722648512620682360432121189510184674501600448851587222090499524990225152460512280
Technical details
Node id
n-294d7f22508b5365 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1dcc13434ed0f3e9

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:     15,      30,       9;
  3:     56,     168,     108,      16;
  4:    210,     840,     840,     280,      25;
  5:    792,    3960,    5400,    2880,     600,      36;
  6:   3003,   18018,   31185,   23100,    7875,    1134,     49;
  7:  11440,   80080,  168168,  160160,   77000,   18480,   1960,    64;
  8:  43758,  350064,  864864, 1009008,  630630,  216216,  38808,  3168,   81;
  9: 167960, 1511640, 4296240, 5940480, 4586400, 2063880, 535080, 74880, 4860, 100;
%F Formula
The row polynomials satisfy (2*n^2+4*n+(n^3+2*n^2)*x)*P(n,x) = (8*n^2+12*n+4+(4*n^3+18*n^2+14*n)*x+(3*n^3+7*n^2+2*n)*x^2)*P(n-1,x) - ((4*n+4)*x+(2*n^2+9*n+6)*x^2+(2*n^2+5*n+2)*x^3)*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^2 + 12*n + 4 + (4*n^3 + 18*n^2 + 14*n)*x + (3*n^3 + 7*n^2 + 2*n)*x^2)*P[n - 1, x] - ((4*n + 4)*x + (2*n^2 + 9*n + 6)*x^2 + (2*n^2 + 5*n + 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 4*n + (n^3 + 2*n^2)*x)]];
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^2 + 12*n + 4 + (4*n^3 + 18*n^2 + 14*n)*x + (3*n^3 + 7*n^2 + 2*n)*x^2)*v[i - 1] - ((4*n + 4)*x + (2*n^2 + 9*n + 6)*x^2 + (2*n^2 + 5*n + 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 4*n + (n^3 + 2*n^2)*x)); 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

Same fingerprint ?

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