Real-rooted atlas

A008459 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + 24\,x + 36\,x^{2} - 4\,n - 24\,n x - 36\,n x^{2}\right)\,P_{n-2}(x) + \left(-4 - 6\,x + 8\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
2226036118
31365406482161540
4886463284246048129621286
559443858094320105840518407776304300
64063631514496876014860801166400419904466564443580
728148825393209407664183405601995840011757312326592027993665830600

All 15 rows

Technical details
Node id
n-37a972deb39d81b2 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-c1efd17c58b17f52

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,         6;
  2:       22,        60,        36;
  3:      136,       540,       648,        216;
  4:      886,      4632,      8424,       6048,       1296;
  5:     5944,     38580,     94320,     105840,      51840,       7776;
  6:    40636,    315144,    968760,    1486080,    1166400,     419904,      46656;
  7:   281488,   2539320,   9407664,   18340560,   19958400,   11757312,    3265920,    279936;
  8:  1968934,  20255280,  87836112,  207918144,  290213280,  239936256,  111041280,  24634368,   1679616;
  9: 13875544, 160320708, 796581216, 2219247072, 3785854464, 4047454656, 2657152512, 997691904, 181398528, 10077696;
%F Formula
The row polynomials satisfy n*P(n,x) = -(4*n-4+(24*n-24)*x+(36*n-36)*x^2)*P(n-2,x) + (8*n-4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 + (24*n - 24)*x + (36*n - 36)*x^2)*P[n - 2, x] + (8*n - 4 + (12*n - 6)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 4 + (24*n - 24)*x + (36*n - 36)*x^2)*v[i - 2] + (8*n - 4 + (12*n - 6)*x)*v[i - 1])/(n)); 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