Real-rooted atlas

A008292 → θ+1 (shift +2) → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 + 2\,x - n - n x - n^{2} x\right)\,P_{n-2}(x) + \left(-2\,x + 2\,x^{2} - n x + n x^{2}\right)\,P_{n-2}'(x) + \left(2 + x + n + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 3 + 8\,x + x^{2}\)
\(P_{2}(x) = 4 + 33\,x + 22\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
138112
243322160
35104198521360
46285120890611412520
57720595596643573240120160
681729257587809562476128794941181440
79401610225652940478095035293643824100411814400
810911738272031863447862124655177018207681435202026119958400

All 15 rows

Technical details
Node id
n-7cab3483238ce3c8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8efdaeff5d010efc

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  3,     8,       1;
   2:  4,    33,      22,        1;
   3:  5,   104,     198,       52,         1;
   4:  6,   285,    1208,      906,       114,          1;
   5:  7,   720,    5955,     9664,      3573,        240,         1;
   6:  8,  1729,   25758,    78095,     62476,      12879,       494,         1;
   7:  9,  4016,  102256,   529404,    780950,     352936,     43824,      1004,        1;
   8: 10,  9117,  382720,  3186344,   7862124,    6551770,   1820768,    143520,     2026,       1;
   9: 11, 20360, 1373733, 17627904,  68166798,   94345488,  48690570,   8813952,   457911,    4072,    1;
  10: 12, 44913, 4782710, 91689165, 530547792, 1137586002, 975073716, 331592370, 40750740, 1434813, 8166, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n+2+(n^2+n-2)*x)*P(n-2,x) - ((n+2)*x-(n+2)*x^2)*d/dx P(n-2,x) + (n+2+(n+1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 3 + 8*x + x^2, P(2,x) = 4 + 33*x + 22*x^2 + x^3.
%t Mathematica
P[1, x_] := 3 + 8*x + x^2;
P[2, x_] := 4 + 33*x + 22*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(n + 2 + (n^2 + n - 2)*x)*P[n - 2, x] - ((n + 2)*x - (n + 2)*x^2)*D[P[n - 2, x], x] + (n + 2 + (n + 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 3 + 8*x + x^2; v[2] = 4 + 33*x + 22*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = -(n + 2 + (n^2 + n - 2)*x)*v[i - 2] - ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 2], x) + (n + 2 + (n + 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.