Real-rooted atlas

A008459 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-n - 2\,n x + n x^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(4 + 6\,x - 2\,x^{3} - 10\,n - 17\,n x - 3\,n x^{2} + 3\,n x^{3} + 2\,n^{2} + 8\,n^{2} x + 6\,n^{2} x^{2} + 4\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2} - 2\,x^{3} - 2\,x^{4} - 3\,n x - 7\,n x^{2} - 4\,n x^{3} - 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2128121
35054151120
4210320150241705
5882175011903303514188
6369690728190336063048124997
71544445276512822835079801092631149488
86435021964830030021067280850168001764801894465
92674101042470167310014294287027022009703234027009915351220
10110853648620008970390906048054654602018016450450580803960120131997493
114585308223554764665089054478710390904801782580851711669309309867056101431191193192
121892909210158220823667243631408520026189163014282611251459408121080961801800160160772216811141524033
137800450045700236411763078601749824076166511345010600375504589339041347746402634231633033002502501037419516809904780
143209328002038517600574633150094753626241014328915673976302403753780030133024320032590958453909856578578037856013650224140592084805
Technical details
Node id
n-dc45df414c12defd · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0ec5637665ef6d37

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:     12,       8,       1;
  3:     50,      54,      15,       1;
  4:    210,     320,     150,      24,      1;
  5:    882,    1750,    1190,     330,     35,      1;
  6:   3696,    9072,    8190,    3360,    630,     48,     1;
  7:  15444,   45276,   51282,   28350,   7980,   1092,    63,    1;
  8:  64350,  219648,  300300,  210672,  80850,  16800,  1764,   80,  1;
  9: 267410, 1042470, 1673100, 1429428, 702702, 200970, 32340, 2700, 99, 1;
%F Formula
The row polynomials satisfy (n^3-n+(2*n^2-2*n)*x+n*x^2)*P(n,x) = (4*n^3+2*n^2-10*n+4+(3*n^3+8*n^2-17*n+6)*x+(6*n^2-3*n)*x^2+(3*n-2)*x^3)*P(n-1,x) - ((2*n^2+3*n-2)*x+(2*n^2+7*n-2)*x^2+(4*n+2)*x^3+2*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[((4*n^3 + 2*n^2 - 10*n + 4 + (3*n^3 + 8*n^2 - 17*n + 6)*x + (6*n^2 - 3*n)*x^2 + (3*n - 2)*x^3)*P[n - 1, x] - ((2*n^2 + 3*n - 2)*x + (2*n^2 + 7*n - 2)*x^2 + (4*n + 2)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 - n + (2*n^2 - 2*n)*x + n*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] = ((4*n^3 + 2*n^2 - 10*n + 4 + (3*n^3 + 8*n^2 - 17*n + 6)*x + (6*n^2 - 3*n)*x^2 + (3*n - 2)*x^3)*v[i - 1] - ((2*n^2 + 3*n - 2)*x + (2*n^2 + 7*n - 2)*x^2 + (4*n + 2)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 - n + (2*n^2 - 2*n)*x + n*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

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. [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