Real-rooted atlas

A116412 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(32 + 14\,x + 2\,x^{2} - 57\,n - 15\,n x + 28\,n^{2} - 2\,n^{2} x - 2\,n^{2} x^{2} - 3\,n^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(52 + 40\,x + 4\,x^{2} - 210\,n - 222\,n x - 66\,n x^{2} - 6\,n x^{3} + 152\,n^{2} + 128\,n^{2} x + 20\,n^{2} x^{2} - 24\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(26 + 46\,x + 22\,x^{2} + 2\,x^{3} - 91\,n - 187\,n x - 123\,n x^{2} - 29\,n x^{3} - 2\,n x^{4} + 39\,n^{2} + 69\,n^{2} x + 33\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
28210
34224369
4190174484416
579710104448052336
6319251783156900120612552
712391245221919475801590168765452

All 15 rows

Technical details
Node id
n-46c2e73c14e9ed58 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2f476f2fd7b79515

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      1;
  2:      8,      2;
  3:     42,     24,      3;
  4:    190,    174,     48,      4;
  5:    797,   1010,    444,     80,      5;
  6:   3192,   5178,   3156,    900,    120,     6;
  7:  12391,  24522,  19194,   7580,   1590,   168,    7;
  8:  47030, 109910, 105288,  53440,  15490,  2562,  224,   8;
  9: 175494, 473094, 536706, 333876, 124335, 28350, 3864, 288, 9;
%F Formula
The row polynomials satisfy -(3*n^3-28*n^2+57*n-32-(3*n^3-2*n^2-15*n+14)*x+(2*n^2-2)*x^2)*P(n,x) = -(24*n^3-152*n^2+210*n-52+(12*n^3-128*n^2+222*n-40)*x-(20*n^2-66*n+4)*x^2+6*n*x^3)*P(n-1,x) + (39*n^2-91*n+26+(69*n^2-187*n+46)*x+(33*n^2-123*n+22)*x^2+(3*n^2-29*n+2)*x^3-2*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 152*n^2 + 210*n - 52 + (12*n^3 - 128*n^2 + 222*n - 40)*x - (20*n^2 - 66*n + 4)*x^2 + 6*n*x^3)*P[n - 1, x] + (39*n^2 - 91*n + 26 + (69*n^2 - 187*n + 46)*x + (33*n^2 - 123*n + 22)*x^2 + (3*n^2 - 29*n + 2)*x^3 - 2*n*x^4)*D[P[n - 1, x], x])/(-3*n^3 + 28*n^2 - 57*n + 32 + (3*n^3 - 2*n^2 - 15*n + 14)*x - (2*n^2 - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n^3 - 152*n^2 + 210*n - 52 + (12*n^3 - 128*n^2 + 222*n - 40)*x - (20*n^2 - 66*n + 4)*x^2 + 6*n*x^3)*v[i - 1] + (39*n^2 - 91*n + 26 + (69*n^2 - 187*n + 46)*x + (33*n^2 - 123*n + 22)*x^2 + (3*n^2 - 29*n + 2)*x^3 - 2*n*x^4)*deriv(v[i - 1], x))/(-3*n^3 + 28*n^2 - 57*n + 32 + (3*n^3 - 2*n^2 - 15*n + 14)*x - (2*n^2 - 2)*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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.