Real-rooted atlas

A008459 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,x + x^{2} - 16\,n + 14\,n x - 2\,n x^{2} + 15\,n^{2} - 10\,n^{2} x + n^{2} x^{2} - 6\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(8\,x - 10\,x^{2} + 2\,x^{3} - 28\,n x + 34\,n x^{2} - 6\,n x^{3} + 3\,n^{2} + 26\,n^{2} x - 26\,n^{2} x^{2} + 3\,n^{2} x^{3} - 2\,n^{3} - 14\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-3\,x + 7\,x^{2} - 5\,x^{3} + x^{4} + 8\,n x - 18\,n x^{2} + 12\,n x^{3} - 2\,n x^{4} - 6\,n^{2} x + 10\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2628
32724354
488168604320
522580060012051750
648628503900162021069072
7931823218375137203675336745276

All 15 rows

Technical details
Node id
n-ae7c075ee4e5b1ad · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-742638bd64a364fd

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:    6,     2;
  3:   27,    24,      3;
  4:   88,   168,     60,      4;
  5:  225,   800,    600,    120,      5;
  6:  486,  2850,   3900,   1620,    210,     6;
  7:  931,  8232,  18375,  13720,   3675,   336,     7;
  8: 1632, 20384,  68208,  82320,  39200,  7392,   504,   8;
  9: 2673, 44928, 211680, 381024, 291060, 96768, 13608, 720, 9;
%F Formula
The row polynomials satisfy (n^4-6*n^3+15*n^2-16*n+6+(2*n^3-10*n^2+14*n-6)*x+(n^2-2*n+1)*x^2)*P(n,x) = (n^4-2*n^3+3*n^2+(3*n^4-14*n^3+26*n^2-28*n+8)*x+(6*n^3-26*n^2+34*n-10)*x^2+(3*n^2-6*n+2)*x^3)*P(n-1,x) + ((2*n^3-6*n^2+8*n-3)*x-(2*n^3-10*n^2+18*n-7)*x^2-(4*n^2-12*n+5)*x^3-(2*n-1)*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[((n^4 - 2*n^3 + 3*n^2 + (3*n^4 - 14*n^3 + 26*n^2 - 28*n + 8)*x + (6*n^3 - 26*n^2 + 34*n - 10)*x^2 + (3*n^2 - 6*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 - 6*n^2 + 8*n - 3)*x - (2*n^3 - 10*n^2 + 18*n - 7)*x^2 - (4*n^2 - 12*n + 5)*x^3 - (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 6*n^3 + 15*n^2 - 16*n + 6 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^2 - 2*n + 1)*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] = ((n^4 - 2*n^3 + 3*n^2 + (3*n^4 - 14*n^3 + 26*n^2 - 28*n + 8)*x + (6*n^3 - 26*n^2 + 34*n - 10)*x^2 + (3*n^2 - 6*n + 2)*x^3)*v[i - 1] + ((2*n^3 - 6*n^2 + 8*n - 3)*x - (2*n^3 - 10*n^2 + 18*n - 7)*x^2 - (4*n^2 - 12*n + 5)*x^3 - (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 6*n^3 + 15*n^2 - 16*n + 6 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^2 - 2*n + 1)*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