Real-rooted atlas

A114687 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 2\,x + 4\,x^{2} + 3\,n + 7\,n x + 2\,n x^{2} + 6\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(-2 - 2\,x + 4\,x^{2} + 3\,n + 5\,n x + 20\,n x^{2} + 12\,n x^{3} + 30\,n^{2} + 48\,n^{2} x + 24\,n^{2} x^{2} + 14\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-4 - 18\,x - 30\,x^{2} - 24\,x^{3} - 8\,x^{4} - 10\,n - 38\,n x - 44\,n x^{2} - 16\,n x^{3} - 4\,n^{2} - 12\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22522451
3129180728389
46811316848208163069
536539070820032805603224795
61982560306707404012011280144064203775
71085453914965668324205601691203584035841281696105

All 15 rows

Technical details
Node id
n-ea661d0f47f9120b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a3dcf688a708825c

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:       5,        2;
  2:      25,       22,        4;
  3:     129,      180,       72,        8;
  4:     681,     1316,      848,      208,       16;
  5:    3653,     9070,     8200,     3280,      560,      32;
  6:   19825,    60306,    70740,    40120,    11280,    1440,      64;
  7:  108545,   391496,   566832,   420560,   169120,   35840,    3584,    128;
  8:  598417,  2498824,  4314688,  3976896,  2106944,  643328,  107520,   8704,   256;
  9: 3317445, 15750234, 31631472, 34948032, 23131584, 9362304, 2268672, 308736, 20736, 512;
%F Formula
The row polynomials satisfy (2*n^3+6*n^2+3*n-2+(4*n^2+7*n-2)*x+(2*n+4)*x^2)*P(n,x) = (14*n^3+30*n^2+3*n-2+(12*n^3+48*n^2+5*n-2)*x+(24*n^2+20*n+4)*x^2+12*n*x^3)*P(n-1,x) - (4*n^2+10*n+4+(12*n^2+38*n+18)*x+(8*n^2+44*n+30)*x^2+(16*n+24)*x^3+8*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[((14*n^3 + 30*n^2 + 3*n - 2 + (12*n^3 + 48*n^2 + 5*n - 2)*x + (24*n^2 + 20*n + 4)*x^2 + 12*n*x^3)*P[n - 1, x] - (4*n^2 + 10*n + 4 + (12*n^2 + 38*n + 18)*x + (8*n^2 + 44*n + 30)*x^2 + (16*n + 24)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 6*n^2 + 3*n - 2 + (4*n^2 + 7*n - 2)*x + (2*n + 4)*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] = ((14*n^3 + 30*n^2 + 3*n - 2 + (12*n^3 + 48*n^2 + 5*n - 2)*x + (24*n^2 + 20*n + 4)*x^2 + 12*n*x^3)*v[i - 1] - (4*n^2 + 10*n + 4 + (12*n^2 + 38*n + 18)*x + (8*n^2 + 44*n + 30)*x^2 + (16*n + 24)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 6*n^2 + 3*n - 2 + (4*n^2 + 7*n - 2)*x + (2*n + 4)*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