Real-rooted atlas

A008288 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12\,x - 6\,x^{2} + 12\,n x - n x^{3} + 6\,n^{2} x^{2} + n^{3} x^{3}\right)\,P_{n}(x) = \left(64 + 96\,x + 40\,x^{2} + 4\,x^{3} - 32\,n + 12\,n x + 54\,n x^{2} + 20\,n x^{3} + 2\,n x^{4} - 24\,n^{2} x - 14\,n^{2} x^{2} + 7\,n^{2} x^{3} + 2\,n^{2} x^{4} - 4\,n^{3} x^{2} - 3\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(32 + 80\,x + 68\,x^{2} + 22\,x^{3} + 2\,x^{4} + 32\,n x + 72\,n x^{2} + 52\,n x^{3} + 13\,n x^{4} + n x^{5} + 8\,n^{2} x^{2} + 16\,n^{2} x^{3} + 9\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3510318
4123624476
529116120445314
6703504922847061272
716910141803142855510275078
84082856614462003320960140820036

All 15 rows

Technical details
Node id
n-fb6fe1c599c2e00f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-06a4c747287235b2

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    2,     2;
   3:    5,    10,     3;
   4:   12,    36,    24,     4;
   5:   29,   116,   120,    44,     5;
   6:   70,   350,   492,   284,    70,     6;
   7:  169,  1014,  1803,  1428,   555,   102,     7;
   8:  408,  2856,  6144,  6200,  3320,   960,   140,    8;
   9:  985,  7880, 19896, 24424, 16730,  6672,  1526,  184,   9;
  10: 2378, 21402, 62040, 89768, 75060, 38364, 12096, 2280, 234, 10;
%F Formula
The row polynomials satisfy ((12*n-12)*x+(6*n^2-6)*x^2+(n^3-n)*x^3)*P(n,x) = -(32*n-64+(24*n^2-12*n-96)*x+(4*n^3+14*n^2-54*n-40)*x^2+(3*n^3-7*n^2-20*n-4)*x^3-(2*n^2+2*n)*x^4)*P(n-1,x) + (32+(32*n+80)*x+(8*n^2+72*n+68)*x^2+(16*n^2+52*n+22)*x^3+(9*n^2+13*n+2)*x^4+(n^2+n)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(32*n - 64 + (24*n^2 - 12*n - 96)*x + (4*n^3 + 14*n^2 - 54*n - 40)*x^2 + (3*n^3 - 7*n^2 - 20*n - 4)*x^3 - (2*n^2 + 2*n)*x^4)*P[n - 1, x] + (32 + (32*n + 80)*x + (8*n^2 + 72*n + 68)*x^2 + (16*n^2 + 52*n + 22)*x^3 + (9*n^2 + 13*n + 2)*x^4 + (n^2 + n)*x^5)*D[P[n - 1, x], x])/((12*n - 12)*x + (6*n^2 - 6)*x^2 + (n^3 - n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(32*n - 64 + (24*n^2 - 12*n - 96)*x + (4*n^3 + 14*n^2 - 54*n - 40)*x^2 + (3*n^3 - 7*n^2 - 20*n - 4)*x^3 - (2*n^2 + 2*n)*x^4)*v[i - 1] + (32 + (32*n + 80)*x + (8*n^2 + 72*n + 68)*x^2 + (16*n^2 + 52*n + 22)*x^3 + (9*n^2 + 13*n + 2)*x^4 + (n^2 + n)*x^5)*deriv(v[i - 1], x))/((12*n - 12)*x + (6*n^2 - 6)*x^2 + (n^3 - n)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.