Real-rooted atlas

A069466 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n^{2} + n^{3} x\right)\,P_{n}(x) = \left(8 + 16\,x + 6\,x^{2} - 32\,n - 52\,n x - 16\,n x^{2} + 32\,n^{2} + 32\,n^{2} x + 2\,n^{2} x^{2} + 16\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 14\,x^{2} + 6\,x^{3} - 16\,n x - 24\,n x^{2} - 8\,n x^{3} - 8\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
2367218126
34001200720802400
449001960018900560035049350
5635043175204233602116803780015121055376
68537765122656873180062092801940400232848646823097228
711778624824503681712430721585584007207200015567552134534427456513042816

All 15 rows

Technical details
Node id
n-2752334636fec784 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-59bab5701f3316b8

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:          4,           4;
  2:         36,          72,          18;
  3:        400,        1200,         720,          80;
  4:       4900,       19600,       18900,        5600,         350;
  5:      63504,      317520,      423360,      211680,       37800,        1512;
  6:     853776,     5122656,     8731800,     6209280,     1940400,      232848,       6468;
  7:   11778624,    82450368,   171243072,   158558400,    72072000,    15567552,    1345344,     27456;
  8:  165636900,  1325095200,  3246483240,  3710266560,  2229727500,   713512800,  113513400,   7413120,   115830;
  9: 2363904400, 21275139600, 60070982400, 81763281600, 61322461200, 26281054800, 6289483200, 770140800, 39382200, 486200;
%F Formula
The row polynomials satisfy (2*n^2+n^3*x)*P(n,x) = (32*n^2-32*n+8+(16*n^3+32*n^2-52*n+16)*x+(12*n^3+2*n^2-16*n+6)*x^2)*P(n-1,x) - ((16*n-8)*x+(8*n^2+24*n-14)*x^2+(8*n^2+8*n-6)*x^3)*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[((32*n^2 - 32*n + 8 + (16*n^3 + 32*n^2 - 52*n + 16)*x + (12*n^3 + 2*n^2 - 16*n + 6)*x^2)*P[n - 1, x] - ((16*n - 8)*x + (8*n^2 + 24*n - 14)*x^2 + (8*n^2 + 8*n - 6)*x^3)*D[P[n - 1, x], x])/(2*n^2 + n^3*x)]];
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] = ((32*n^2 - 32*n + 8 + (16*n^3 + 32*n^2 - 52*n + 16)*x + (12*n^3 + 2*n^2 - 16*n + 6)*x^2)*v[i - 1] - ((16*n - 8)*x + (8*n^2 + 24*n - 14)*x^2 + (8*n^2 + 8*n - 6)*x^3)*deriv(v[i - 1], x))/(2*n^2 + n^3*x)); 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