Real-rooted atlas

A110441 → basis A013609 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
216824
3889624208
4416720384641584
5181643523648128016011256
67552232322688014720384038476608
7304001144321699201280005184010752896506240
81195525328649692169356805196801666562867220483274368
94619842380800513561660712964314240189235250176073728460820836384
10176102410305408257464323610880031395840175902726365184143769618432010240130905216
116639744435041281236307202009497602072268801419264006550118420152320396288045056022528813967104
122481100817996467257354240010612633601268490240102986956858043596822745088060825600105881601081344491525018372352
139202163273209446425868021765372600320731266560068800020484593770496219620966474714112017661952027574272255590410649630720163712
14339144704293653504011396407296262660136964014518272043010666496332675481601882998374478083850242345615360496697344702873605963776229376186918660096
Technical details
Node id
n-3fedd939d800e947 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-226ea59bdb20c50c

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:      2;
  2:     16,       8;
  3:     88,      96,      24;
  4:    416,     720,     384,      64;
  5:   1816,    4352,    3648,    1280,     160;
  6:   7552,   23232,   26880,   14720,    3840,     384;
  7:  30400,  114432,  169920,  128000,   51840,   10752,    896;
  8: 119552,  532864,  969216,  935680,  519680,  166656,  28672,  2048;
  9: 461984, 2380800, 5135616, 6071296, 4314240, 1892352, 501760, 73728, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+4+(n+2)*x)*P(n-1,x) + (2+4*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (n + 2)*x)*P[n - 1, x] + (2 + 4*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 4 + (n + 2)*x)*v[i - 1] + (2 + 4*x + x^2)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023