Real-rooted atlas

A108838 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-18 + 15\,x - 3\,x^{2} + 21\,n - 16\,n x + 2\,n x^{2} - 3\,n^{2} - n^{2} x + n^{2} x^{2} - n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(24 - 42\,x + 22\,x^{2} - 4\,x^{3} + 14\,n - 4\,n x - 7\,n x^{2} + n x^{3} + 5\,n^{2} + 11\,n^{2} x - 9\,n^{2} x^{2} + 3\,n^{2} x^{3} + 4\,n^{3} - 4\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(6\,x - 12\,x^{2} + 8\,x^{3} - 2\,x^{4} + 5\,n x - 7\,n x^{2} + 4\,n x^{3} - 2\,n x^{4} - n^{2} x + 5\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
212416
35042698
4160264968528
54201190840180102640
69524200504020803001212584
7193212348229321596044104621458058
8360031584846729072042000840067216261664
96270724682661124102562910609702014784936181158924
101034015246073656015523201580040803880203280244801260205064640
1116302298870184041050965207078500515314819819803946803861016502221900692
122475255255242282241491204027181440269549281479878444684647207205852021122493902560
13364009724269057048396710609202050011962665089321232384143769369360125125085800265226399828780
145208016409121828444897441344280560280463783320453477024264840576918197281849848020820801223043276281692605880
Technical details
Node id
n-555b12ad087e7af5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5aba915c432f3abc

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:    2;
  2:   12,     4;
  3:   50,    42,      6;
  4:  160,   264,     96,      8;
  5:  420,  1190,    840,    180,     10;
  6:  952,  4200,   5040,   2080,    300,    12;
  7: 1932, 12348,  22932,  15960,   4410,   462,    14;
  8: 3600, 31584,  84672,  90720,  42000,  8400,   672,  16;
  9: 6270, 72468, 266112, 410256, 291060, 97020, 14784, 936, 18;
%F Formula
The row polynomials satisfy (n^4-n^3-3*n^2+21*n-18+(2*n^3-n^2-16*n+15)*x+(n^2+2*n-3)*x^2)*P(n,x) = (n^4+4*n^3+5*n^2+14*n+24+(3*n^4-4*n^3+11*n^2-4*n-42)*x+(6*n^3-9*n^2-7*n+22)*x^2+(3*n^2+n-4)*x^3)*P(n-1,x) + ((2*n^3-n^2+5*n+6)*x-(2*n^3-5*n^2+7*n+12)*x^2-(4*n^2-4*n-8)*x^3-(2*n+2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 4*n^3 + 5*n^2 + 14*n + 24 + (3*n^4 - 4*n^3 + 11*n^2 - 4*n - 42)*x + (6*n^3 - 9*n^2 - 7*n + 22)*x^2 + (3*n^2 + n - 4)*x^3)*P[n - 1, x] + ((2*n^3 - n^2 + 5*n + 6)*x - (2*n^3 - 5*n^2 + 7*n + 12)*x^2 - (4*n^2 - 4*n - 8)*x^3 - (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 - n^3 - 3*n^2 + 21*n - 18 + (2*n^3 - n^2 - 16*n + 15)*x + (n^2 + 2*n - 3)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 + 4*n^3 + 5*n^2 + 14*n + 24 + (3*n^4 - 4*n^3 + 11*n^2 - 4*n - 42)*x + (6*n^3 - 9*n^2 - 7*n + 22)*x^2 + (3*n^2 + n - 4)*x^3)*v[i - 1] + ((2*n^3 - n^2 + 5*n + 6)*x - (2*n^3 - 5*n^2 + 7*n + 12)*x^2 - (4*n^2 - 4*n - 8)*x^3 - (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 - n^3 - 3*n^2 + 21*n - 18 + (2*n^3 - n^2 - 16*n + 15)*x + (n^2 + 2*n - 3)*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