Real-rooted atlas

A367270 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-n + 2\,n x + 5\,n^{2} - 14\,n^{2} x - 8\,n^{3} + 36\,n^{3} x - 4\,n^{3} x^{2} + 4\,n^{4} - 36\,n^{4} x + 12\,n^{4} x^{2} + 12\,n^{5} x - 12\,n^{5} x^{2} + 4\,n^{6} x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 24\,x^{2} + 16\,x^{3} - 14\,n - 68\,n x - 108\,n x^{2} - 40\,n x^{3} + 36\,n^{2} + 120\,n^{2} x + 88\,n^{2} x^{2} - 40\,n^{3} - 40\,n^{3} x + 136\,n^{3} x^{2} + 72\,n^{3} x^{3} + 16\,n^{4} - 72\,n^{4} x - 220\,n^{4} x^{2} - 48\,n^{4} x^{3} + 48\,n^{5} x + 64\,n^{5} x^{2} - 24\,n^{5} x^{3} + 16\,n^{6} x^{2} + 24\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 12\,x^{3} + 8\,x^{4} - 6\,n x - 28\,n x^{2} - 40\,n x^{3} - 16\,n x^{4} + 12\,n^{2} x + 32\,n^{2} x^{2} + 12\,n^{2} x^{3} - 8\,n^{2} x^{4} - 8\,n^{3} x + 8\,n^{3} x^{2} + 64\,n^{3} x^{3} + 32\,n^{3} x^{4} - 24\,n^{4} x^{2} - 52\,n^{4} x^{3} - 8\,n^{4} x^{4} - 8\,n^{5} x^{3} - 16\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
231619
31096108214
43548010805122107
5126224075607680200019606
6462100804536071680420006912176494
7171644352249480537600504000193536219521552636
8643519219212972963548160462000029030407902726553613422931
9243108236806486480215255043603600031933440144883202949120186624114453478
10923783500640315057601230028802522520002905943041883481606488064010264320512000964953082
11352716147804801496523606721228801633632000232475443219776556809840230402668723203379200013629448059000852
121352078620780166983776803547315200997682400016937496576179307448321180827648046702656001025024000106309632353894466767603038
13520030025959897632125373281820955136058198140000114933012480146007493632120444420096635156121602050048000037208371203220439048998912549337926268
142005830010816624001460244240091378839552327202876000737486830080109505620224010897352294407240779786243136573440008433897472012881756160944885760224788484492487558524
Technical details
Node id
n-eba6e6c2bf83470e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b9c4940a2f76c6c3

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:     1;
  2:     3,     16;
  3:    10,     96,     108;
  4:    35,    480,    1080,      512;
  5:   126,   2240,    7560,     7680,     2000;
  6:   462,  10080,   45360,    71680,    42000,     6912;
  7:  1716,  44352,  249480,   537600,   504000,   193536,    21952;
  8:  6435, 192192, 1297296,  3548160,  4620000,  2903040,   790272,   65536;
  9: 24310, 823680, 6486480, 21525504, 36036000, 31933440, 14488320, 2949120, 186624;
%F Formula
The row polynomials satisfy (4*n^4-8*n^3+5*n^2-n+(12*n^5-36*n^4+36*n^3-14*n^2+2*n)*x+(4*n^6-12*n^5+12*n^4-4*n^3)*x^2)*P(n,x) = (16*n^4-40*n^3+36*n^2-14*n+2+(48*n^5-72*n^4-40*n^3+120*n^2-68*n+12)*x+(16*n^6+64*n^5-220*n^4+136*n^3+88*n^2-108*n+24)*x^2+(24*n^6-24*n^5-48*n^4+72*n^3-40*n+16)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+6*n-1)*x+(24*n^4-8*n^3-32*n^2+28*n-6)*x^2+(8*n^5+52*n^4-64*n^3-12*n^2+40*n-12)*x^3+(16*n^5+8*n^4-32*n^3+8*n^2+16*n-8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (48*n^5 - 72*n^4 - 40*n^3 + 120*n^2 - 68*n + 12)*x + (16*n^6 + 64*n^5 - 220*n^4 + 136*n^3 + 88*n^2 - 108*n + 24)*x^2 + (24*n^6 - 24*n^5 - 48*n^4 + 72*n^3 - 40*n + 16)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (24*n^4 - 8*n^3 - 32*n^2 + 28*n - 6)*x^2 + (8*n^5 + 52*n^4 - 64*n^3 - 12*n^2 + 40*n - 12)*x^3 + (16*n^5 + 8*n^4 - 32*n^3 + 8*n^2 + 16*n - 8)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 8*n^3 + 5*n^2 - n + (12*n^5 - 36*n^4 + 36*n^3 - 14*n^2 + 2*n)*x + (4*n^6 - 12*n^5 + 12*n^4 - 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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (48*n^5 - 72*n^4 - 40*n^3 + 120*n^2 - 68*n + 12)*x + (16*n^6 + 64*n^5 - 220*n^4 + 136*n^3 + 88*n^2 - 108*n + 24)*x^2 + (24*n^6 - 24*n^5 - 48*n^4 + 72*n^3 - 40*n + 16)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (24*n^4 - 8*n^3 - 32*n^2 + 28*n - 6)*x^2 + (8*n^5 + 52*n^4 - 64*n^3 - 12*n^2 + 40*n - 12)*x^3 + (16*n^5 + 8*n^4 - 32*n^3 + 8*n^2 + 16*n - 8)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 8*n^3 + 5*n^2 - n + (12*n^5 - 36*n^4 + 36*n^3 - 14*n^2 + 2*n)*x + (4*n^6 - 12*n^5 + 12*n^4 - 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