Real-rooted atlas

A133214 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(8 + 4\,x + 16\,n + 12\,n x + 7\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-3 - 9\,x - 6\,x^{2} - 2\,n - 6\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 16 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
016824
111412024258
27681248576642656
35010110408160224016026610
4320168916091200412807680384261720
5201698679224889560566720178080241928962540370
61257472496832079403526549760293888068812871680204824416640
77777314352707846658444867560192396345601341849624514562027524608232904610

All 15 rows

Technical details
Node id
n-e5037b63434d4540 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3ff6c47fdabd4f1c

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:        16,          8;
  1:       114,        120,         24;
  2:       768,       1248,        576,         64;
  3:      5010,      11040,       8160,       2240,        160;
  4:     32016,      89160,      91200,      41280,       7680,        384;
  5:    201698,     679224,     889560,     566720,     178080,      24192,       896;
  6:   1257472,    4968320,    7940352,    6549760,    2938880,     688128,     71680,      2048;
  7:   7777314,   35270784,   66584448,   67560192,   39634560,   13418496,   2451456,    202752,     4608;
  8:  47800080,  244712520,  533018880,  642673920,  467066880,  208535040,  55695360,   8202240,   552960,   10240;
  9: 292292946, 1667290680, 4116519000, 5753552640, 4995661440, 2780692992, 986979840, 214579200, 26104320, 1464320, 22528;
%F Formula
The row polynomials satisfy (n^2+2*n+1)*P(n,x) = (7*n^2+16*n+8+(6*n^2+12*n+4)*x)*P(n-1,x) - (2*n+3+(6*n+9)*x+(4*n+6)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 16 + 8*x.
%t Mathematica
P[0, x_] := 16 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^2 + 16*n + 8 + (6*n^2 + 12*n + 4)*x)*P[n - 1, x] - (2*n + 3 + (6*n + 9)*x + (4*n + 6)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 16 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^2 + 16*n + 8 + (6*n^2 + 12*n + 4)*x)*v[i - 1] - (2*n + 3 + (6*n + 9)*x + (4*n + 6)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n + 1)); 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