Real-rooted atlas

A110441 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-4 - 5\,x - x^{2} + 12\,n + 11\,n x + n x^{2}\right)\,P_{n-1}(x) + \left(-8 - 16\,x - 9\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
21420741
3481006815231
4164448450196311289
5560189225121632516637175
61912769612704109765214128412739913
7652830496601926492041200153243076255221991
8222881185282722403520962798201395204245871725111234633
97609645387211891201793504171310410722524385441126001638810236866503
10259808171756850549288711168972042473258083768296130281628872636868204738188457
1188704064375682102649640758224520367364588057628458304123922723703088720852819244095212387175
12302854423937024859189441850352002660262402686768641951942721026135683866386010158400176123418022881911181202569
131034009688410432345909504819400064131005120014913616321242295296765658176348604272115586580270667044227024393220163836569320583
1435303296324673792137522624035537085446254373824792407104074461176005266160640281322108011279597603335427687038601699941908519723276736535623529
Technical details
Node id
n-4450f75604de5099 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d0d4468560fae926

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,      3;
  2:    14,     20,       7;
  3:    48,    100,      68,      15;
  4:   164,    448,     450,     196,      31;
  5:   560,   1892,    2512,    1632,     516,      63;
  6:  1912,   7696,   12704,   10976,    5214,    1284,    127;
  7:  6528,  30496,   60192,   64920,   41200,   15324,   3076,    255;
  8: 22288, 118528,  272240,  352096,  279820,  139520,  42458,   7172,   511;
  9: 76096, 453872, 1189120, 1793504, 1713104, 1072252, 438544, 112600, 16388, 1023;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (12*n-4+(11*n-5)*x+(n-1)*x^2)*P(n-1,x) - (8+16*x+9*x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((12*n - 4 + (11*n - 5)*x + (n - 1)*x^2)*P[n - 1, x] - (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], x])/(2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((12*n - 4 + (11*n - 5)*x + (n - 1)*x^2)*v[i - 1] - (8 + 16*x + 9*x^2 + x^3)*deriv(v[i - 1], x))/(2*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

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023