Real-rooted atlas

A104684 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2235
34161333
486011463245
5161926247683211921
632560272055205010168315525
7641536103203040044580320168989127905
8128403235616141680296520339612201698486391067925
92561024011468858777616374402646784248416012574722657299004545
1051225344350208223641679269121689004822194816176353927777314146256376499525
11102461440102528079564803475584093413376160668480177672960122356260478000808097453653808673
12204814643229004802682240014099712046344883299913228814383881601372173000833645340292292946450467195614995765
13409634406479749128650752053716608021100400645531344896993261772812283341840102988617605596799472177985612825159596948416454529
14819279872021405696268849152194278656089580142082791726080060686106624932474537281010041375207551873014437119747600107999423221409933619418895174885
15163841835008562831368095825926725182464358799400961306282897923361993789446231285540488359906114568052206242085431437298562436787527726533647310479238482533634723102113
Technical details
Node id
n-83a30b01f0214b4b · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e7d8403424fc1d9b

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   2,     3;
   3:   4,    16,     13;
   4:   8,    60,    114,      63;
   5:  16,   192,    624,     768,     321;
   6:  32,   560,   2720,    5520,    5010,     1683;
   7:  64,  1536,  10320,   30400,   44580,    32016,     8989;
   8: 128,  4032,  35616,  141680,  296520,   339612,   201698,    48639;
   9: 256, 10240, 114688,  587776, 1637440,  2646784,  2484160,  1257472,  265729;
  10: 512, 25344, 350208, 2236416, 7926912, 16890048, 22194816, 17635392, 7777314, 1462563;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n+1)*x-(2*n-4)*x^2)*P(n-1,x) + (4*x+6*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n + 1)*x - (2*n - 4)*x^2)*P[n - 1, x] + (4*x + 6*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 2 + (n + 1)*x - (2*n - 4)*x^2)*v[i - 1] + (4*x + 6*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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