Real-rooted atlas

A114656 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2224
3312419
443648896
558024016016501
661508001200480322668
7725221005600504013446414407
883924704196003136018816358412878592
99576940856448141120150528645129216256432073
101081017280141120508032846720645120207360230405122390004
1111110029700316800155232037255684435200253440063360056320102413286043
1212145248400653400418176013660416234178562090880092928001858560135168204874160672
131318727550412584001019304043490304101477376130470912906048003221504052715523194884096415382397
141423661135682290288229028801236755523769159686596029446596029443664460801066024961453670474547281922333445468
15152940165620397488048096048320640320123675552028268697603847683840307814707213991577603391897603913728017203201638413141557519
Technical details
Node id
n-69ce0ad43c657483 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c5faa49c64af63fa

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,   2;
   3:  3,  12,     4;
   4:  4,  36,    48,      8;
   5:  5,  80,   240,    160,     16;
   6:  6, 150,   800,   1200,    480,     32;
   7:  7, 252,  2100,   5600,   5040,   1344,     64;
   8:  8, 392,  4704,  19600,  31360,  18816,   3584,    128;
   9:  9, 576,  9408,  56448, 141120, 150528,  64512,   9216,   256;
  10: 10, 810, 17280, 141120, 508032, 846720, 645120, 207360, 23040, 512;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (n^2+(6*n^2-12*n+4)*x)*P(n-1,x) + ((2*n-1)*x-(4*n-2)*x^2)*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[((n^2 + (6*n^2 - 12*n + 4)*x)*P[n - 1, x] + ((2*n - 1)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
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] = ((n^2 + (6*n^2 - 12*n + 4)*x)*v[i - 1] + ((2*n - 1)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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