Real-rooted atlas

A126216 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 8\,x + 4\,n + 12\,n x + 2\,n x^{2} + 4\,n^{2} x + 3\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(8 + 8\,x + 28\,n + 56\,n x + 26\,n x^{2} + 2\,n x^{3} + 32\,n^{2} x + 33\,n^{2} x^{2} + 7\,n^{2} x^{3} + 7\,n^{3} x^{2} + 3\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 28\,x - 32\,x^{2} - 14\,x^{3} - 2\,x^{4} - 16\,n x - 34\,n x^{2} - 23\,n x^{3} - 5\,n x^{4} - 4\,n^{2} x^{2} - 6\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21114328
34584364169
41974843127251070
59032750237084012566994
64279155461678580601875198746750
72079387752113904691602205036962948317657
810304949536875129655216022442052080666441692185462
95188592798586485686841926082075220618408110544112325671015182902
10264672315827910309349353068412017923500654040815111602162401795575011106303712
111364886989625756194806260218343180147105750636229441809099633660003964952750096812749034730
1271039373508129452121582058415197226001160574525580962888196569912452168646963165689700406561224135305730956
1337269351928842637987533676566103908090008872365645504868735819835055245449055041041813631355783011482905834415211437749854276
1419688015191639046526646406380293700116460686610883779542162318198188799060456040440120138391753522447425025094069184220481627186215269604206866
Technical details
Node id
n-e75dc83934fd9c75 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-96b6926a4c90aeb1

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:      3,       2;
  2:     11,      14,       3;
  3:     45,      84,      36,       4;
  4:    197,     484,     312,      72,       5;
  5:    903,    2750,    2370,     840,     125,      6;
  6:   4279,   15546,   16785,    8060,    1875,    198,      7;
  7:  20793,   87752,  113904,   69160,   22050,   3696,    294,     8;
  8: 103049,  495368,  751296,  552160,  224420,  52080,   6664,   416,   9;
  9: 518859, 2798586, 4856868, 4192608, 2075220, 618408, 110544, 11232, 567, 10;
%F Formula
The row polynomials satisfy (4*n+8+(4*n^2+12*n+8)*x+(n^3+3*n^2+2*n)*x^2)*P(n,x) = (28*n+8+(32*n^2+56*n+8)*x+(7*n^3+33*n^2+26*n)*x^2+(3*n^3+7*n^2+2*n)*x^3)*P(n-1,x) - (8+(16*n+28)*x+(4*n^2+34*n+32)*x^2+(6*n^2+23*n+14)*x^3+(2*n^2+5*n+2)*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[((28*n + 8 + (32*n^2 + 56*n + 8)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (3*n^3 + 7*n^2 + 2*n)*x^3)*P[n - 1, x] - (8 + (16*n + 28)*x + (4*n^2 + 34*n + 32)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (2*n^2 + 5*n + 2)*x^4)*D[P[n - 1, x], x])/(4*n + 8 + (4*n^2 + 12*n + 8)*x + (n^3 + 3*n^2 + 2*n)*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] = ((28*n + 8 + (32*n^2 + 56*n + 8)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (3*n^3 + 7*n^2 + 2*n)*x^3)*v[i - 1] - (8 + (16*n + 28)*x + (4*n^2 + 34*n + 32)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (2*n^2 + 5*n + 2)*x^4)*deriv(v[i - 1], x))/(4*n + 8 + (4*n^2 + 12*n + 8)*x + (n^3 + 3*n^2 + 2*n)*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