Real-rooted atlas

A125662 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4 + 2\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(3 + 4\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
216824
3929624212
4464744384641656
5218247363744128016012102
69824269282880015040384038484816
7429361422721922401356805280010752896577576
8183648713904116697610403205465601693442867220483851472
97727463447936661737671375364734240197836850892873728460825275466
10320996816168824356463364515616036234240191063046623232145612818432010240163795752
11131965647406723218447444026854400025309312016140902470565376208896004008960450560225281050721404
125379140833288636092445312015209516801647498240123395328065225932824339456062853120107008001081344491526683872392
132177001101472643840451199712082804710401013927904087283998725417696256244356710479504128018202624027844608255590410649642219328910
14875718080642848944021541466880436329248005961079552058013915904413847920642191417958486153379842484162560510754816709263365963776229376265089657120
Technical details
Node id
n-b3e46639d5f3e1f0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c3069d642de4e651

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:     92,      96,      24;
  4:    464,     744,     384,      64;
  5:   2182,    4736,    3744,    1280,     160;
  6:   9824,   26928,   28800,   15040,    3840,     384;
  7:  42936,  142272,  192240,  135680,   52800,   10752,    896;
  8: 183648,  713904, 1166976, 1040320,  546560,  169344,  28672,  2048;
  9: 772746, 3447936, 6617376, 7137536, 4734240, 1978368, 508928, 73728, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+4+(n+2)*x)*P(n-1,x) + (3+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] + (3 + 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] + (3 + 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