Real-rooted atlas

A091044 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
212321660
33216019264448
480640134410242563344
51922240716892165120102424960
64487168322566144056320245764096186304
7102421504129024337920450560319488114688163841390592
823046144047308816220162928640298188817203205242886553610379520
95120168960162201670287361640038422364160183500808912896235929626214477473792
10112644505605271552281149448200192014313062415597568010695475244826624104857601048576578272256
1124576117145616400384105431040374865920811073536112302489610160701445976883202202009604613734441943044316282880
125324829818884920115237486592015931801604171235328711249100881285611526275727360322961408010611589122013265921677721632217174016
13114688745472014313062412745441286372720640198133678084064280576056899928064552264007683714056192016978542592503316480087241523267108864240472260608
142457601835008040553676841712353282421633843288059412480213374730240357656690688423402405888356549394432212231782400872415232002355521126437580963842684354561794909388800
Technical details
Node id
n-8fd5934c6df733bc · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d6d34c33242db792

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,      4;
  2:   12,     32,      16;
  3:   32,    160,     192,      64;
  4:   80,    640,    1344,    1024,      256;
  5:  192,   2240,    7168,    9216,     5120,     1024;
  6:  448,   7168,   32256,   61440,    56320,    24576,     4096;
  7: 1024,  21504,  129024,  337920,   450560,   319488,   114688,   16384;
  8: 2304,  61440,  473088, 1622016,  2928640,  2981888,  1720320,  524288,   65536;
  9: 5120, 168960, 1622016, 7028736, 16400384, 22364160, 18350080, 8912896, 2359296, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+2+(2*n+2)*x)*P(n-1,x) + (4*x+2*x^2)*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[((2*n + 2 + (2*n + 2)*x)*P[n - 1, x] + (4*x + 2*x^2)*D[P[n - 1, x], x])/(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] = ((2*n + 2 + (2*n + 2)*x)*v[i - 1] + (4*x + 2*x^2)*deriv(v[i - 1], x))/(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