Real-rooted atlas

A033282 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 2\,x - 2\,x^{2} + 4\,n + n x - n x^{2} + 3\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(16 + 10\,x - 6\,x^{2} - 4\,x^{3} + 26\,n + 17\,n x - 3\,n x^{2} - 2\,n x^{3} + 15\,n^{2} + 10\,n^{2} x + 3\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(4 + 2\,x - 8\,x^{2} - 8\,x^{3} - 2\,x^{4} + 8\,n + 12\,n x + 4\,n x^{2} + 4\,n^{2} + 6\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22625556
313821910514476
474316661316420424187
5404311780131406900165013237645
6221807973111556085635336606435429343630
71224685244059378609028255005001571572502514303171670

All 15 rows

Technical details
Node id
n-22512e1f0b3f785c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fc86a5a5e40346c1

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:      26,       25,        5;
  3:     138,      219,      105,       14;
  4:     743,     1666,     1316,      420,       42;
  5:    4043,    11780,    13140,     6900,     1650,      132;
  6:   22180,    79731,   115560,    85635,    33660,     6435,      429;
  7:  122468,   524405,   937860,   902825,   500500,   157157,    25025,    1430;
  8:  679757,  3380740,  7203812,  8556548,  6160154,  2726724,   712712,   97240,   4862;
  9: 3789297, 21477510, 53169012, 75248628, 66835314, 38407824, 14145768, 3166488, 377910, 16796;
%F Formula
The row polynomials satisfy (n^3+3*n^2+4*n+4+(n+2)*x-(n+2)*x^2)*P(n,x) = (3*n^3+15*n^2+26*n+16+(2*n^3+10*n^2+17*n+10)*x-(3*n+6)*x^2-(2*n+4)*x^3)*P(n-1,x) + (4*n^2+8*n+4+(6*n^2+12*n+2)*x+(2*n^2+4*n-8)*x^2-8*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[((3*n^3 + 15*n^2 + 26*n + 16 + (2*n^3 + 10*n^2 + 17*n + 10)*x - (3*n + 6)*x^2 - (2*n + 4)*x^3)*P[n - 1, x] + (4*n^2 + 8*n + 4 + (6*n^2 + 12*n + 2)*x + (2*n^2 + 4*n - 8)*x^2 - 8*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 4*n + 4 + (n + 2)*x - (n + 2)*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] = ((3*n^3 + 15*n^2 + 26*n + 16 + (2*n^3 + 10*n^2 + 17*n + 10)*x - (3*n + 6)*x^2 - (2*n + 4)*x^3)*v[i - 1] + (4*n^2 + 8*n + 4 + (6*n^2 + 12*n + 2)*x + (2*n^2 + 4*n - 8)*x^2 - 8*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 4*n + 4 + (n + 2)*x - (n + 2)*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