Real-rooted atlas

A033282 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n + n^{2}\right)\,P_{n}(x) = \left(6\,x + 3\,n + 8\,n x + n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2} + 2\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2103040
318126168312
42833610088402212
54072036006600396014920
6541350990029700386101801897632
77023102310010010021021021021080080626080
888369648048280280840840134534410890883500643957448
91085616917286879602751840623750480196485441904151164024747948
10130819016380015288007796880233906404232592045349200264537006466460153483720
11154115502772003141600197920807520990417907120026860680024622290012584726027457584945638232
12180158404488006058800460468802148854406446563201266289200161803620012944289605883768001158924005795135820
1320821216700128110853609976824055870214420485745285048272944841378824093486536006629045280271188216048674808035357242128
1423827846105814819399380203693490134437703458896517681766895530436810323550531704673505220373158033220556460123512325302035917450214919392128
Technical details
Node id
n-8ce6f1c60ab8d411 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-033fc5829f38f9ac

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:   4;
  2:  10,   30;
  3:  18,  126,   168;
  4:  28,  336,  1008,    840;
  5:  40,  720,  3600,   6600,    3960;
  6:  54, 1350,  9900,  29700,   38610,   18018;
  7:  70, 2310, 23100, 100100,  210210,  210210,   80080;
  8:  88, 3696, 48048, 280280,  840840, 1345344, 1089088,  350064;
  9: 108, 5616, 91728, 687960, 2751840, 6237504, 8019648, 5441904, 1511640;
%F Formula
The row polynomials satisfy (n^2+n-2)*P(n,x) = (n^2+3*n+(2*n^2+8*n+6)*x)*P(n-1,x) + ((2*n+2)*x+(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n + (2*n^2 + 8*n + 6)*x)*P[n - 1, x] + ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n + (2*n^2 + 8*n + 6)*x)*v[i - 1] + ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + n - 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

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