Real-rooted atlas

A143685 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(20\,n + 20\,n x + 19\,n x^{2} + 38\,n x^{3} + n x^{4} + 20\,n^{2} + 30\,n^{2} x + 39\,n^{2} x^{2} + n^{2} x^{3} + 10\,n^{3} x\right)\,P_{n}(x) = \left(-360 + 40\,x + 1218\,x^{2} + 877\,x^{3} + 60\,x^{4} + x^{5} + 660\,n + 1080\,n x - 253\,n x^{2} - 683\,n x^{3} - 18\,n x^{4} + 20\,n^{2} - 710\,n^{2} x - 731\,n^{2} x^{2} - 19\,n^{2} x^{3} - 200\,n^{3} - 190\,n^{3} x\right)\,P_{n-1}(x) + \left(800 + 3960\,x + 7140\,x^{2} + 5679\,x^{3} + 1779\,x^{4} + 81\,x^{5} + x^{6} + 1200\,n + 3960\,n x + 4390\,n x^{2} + 1709\,n x^{3} + 80\,n x^{4} + n x^{5} + 400\,n^{2} + 800\,n^{2} x + 410\,n^{2} x^{2} + 10\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
22615142
3110114271252
46158903453911890
528215415330571951112312
61382832901270009049123663184078
76375618226019194691374198161896751551124

All 15 rows

Technical details
Node id
n-f705eeeb914f0955 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e13294c857653aa6

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,       1;
  2:      26,      15,       1;
  3:     110,     114,      27,       1;
  4:     615,     890,     345,      39,       1;
  5:    2821,    5415,    3305,     719,      51,      1;
  6:   13828,   32901,   27000,    9049,    1236,     63,     1;
  7:   63756,  182260,  191946,   91374,   19816,   1896,    75,    1;
  8:  297245,  991764, 1272982,  788210,  244506,  37300,  2699,   87,  1;
  9: 1355255, 5179005, 7879590, 6074778, 2511702, 552330, 63195, 3645, 99, 1;
%F Formula
The row polynomials satisfy (20*n^2+20*n+(10*n^3+30*n^2+20*n)*x+(39*n^2+19*n)*x^2+(n^2+38*n)*x^3+n*x^4)*P(n,x) = -(200*n^3-20*n^2-660*n+360+(190*n^3+710*n^2-1080*n-40)*x+(731*n^2+253*n-1218)*x^2+(19*n^2+683*n-877)*x^3+(18*n-60)*x^4-x^5)*P(n-1,x) + (400*n^2+1200*n+800+(800*n^2+3960*n+3960)*x+(410*n^2+4390*n+7140)*x^2+(10*n^2+1709*n+5679)*x^3+(80*n+1779)*x^4+(n+81)*x^5+x^6)*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[(-(200*n^3 - 20*n^2 - 660*n + 360 + (190*n^3 + 710*n^2 - 1080*n - 40)*x + (731*n^2 + 253*n - 1218)*x^2 + (19*n^2 + 683*n - 877)*x^3 + (18*n - 60)*x^4 - x^5)*P[n - 1, x] + (400*n^2 + 1200*n + 800 + (800*n^2 + 3960*n + 3960)*x + (410*n^2 + 4390*n + 7140)*x^2 + (10*n^2 + 1709*n + 5679)*x^3 + (80*n + 1779)*x^4 + (n + 81)*x^5 + x^6)*D[P[n - 1, x], x])/(20*n^2 + 20*n + (10*n^3 + 30*n^2 + 20*n)*x + (39*n^2 + 19*n)*x^2 + (n^2 + 38*n)*x^3 + n*x^4)]];
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] = (-(200*n^3 - 20*n^2 - 660*n + 360 + (190*n^3 + 710*n^2 - 1080*n - 40)*x + (731*n^2 + 253*n - 1218)*x^2 + (19*n^2 + 683*n - 877)*x^3 + (18*n - 60)*x^4 - x^5)*v[i - 1] + (400*n^2 + 1200*n + 800 + (800*n^2 + 3960*n + 3960)*x + (410*n^2 + 4390*n + 7140)*x^2 + (10*n^2 + 1709*n + 5679)*x^3 + (80*n + 1779)*x^4 + (n + 81)*x^5 + x^6)*deriv(v[i - 1], x))/(20*n^2 + 20*n + (10*n^3 + 30*n^2 + 20*n)*x + (39*n^2 + 19*n)*x^2 + (n^2 + 38*n)*x^3 + n*x^4)); 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.