Real-rooted atlas

A091044 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(3 + 2\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(6\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
227481691
310836028864820
44052160302415362567381
514581134024192207367680102466430
6510354432163296207360126720368644096597871
71749624494497977617107201520640718848172032163845380840
8590491049760538876812317184148262401006387238707207864326553648427561
91968304330260277136648006169612454041611321856061931520200540163538944262144435848050
106495391732104013510411248037017693405312010868981767896268803609722881008599041572864010485763922632451
1121257646755205663048585627020822406404935680923863449685279703045143855104201719808049545216069206016419430435303692060
126908733257926032283718635214411105280408314649607126946611281015717888617262612483177086976010899947520238760755230198988816777216317733228541
1322320522967222620123804495367349663692824498878976050779494604869442043904064812574310441937548083218802409472057302581248113246208001308622848671088642859599056870
1471744535357128352052616910528360801672192139643610163233852996403205468560957440611089986355248228180295682707546963968107442339840029444014080052999225344563714457626843545625736391511831
Technical details
Node id
n-39cdbb608deef22e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-93af6d1da209b590

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:      6,       4;
  2:     27,      48,       16;
  3:    108,     360,      288,       64;
  4:    405,    2160,     3024,     1536,       256;
  5:   1458,   11340,    24192,    20736,      7680,      1024;
  6:   5103,   54432,   163296,   207360,    126720,     36864,     4096;
  7:  17496,  244944,   979776,  1710720,   1520640,    718848,   172032,    16384;
  8:  59049, 1049760,  5388768, 12317184,  14826240,  10063872,  3870720,   786432,   65536;
  9: 196830, 4330260, 27713664, 80061696, 124540416, 113218560, 61931520, 20054016, 3538944, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n+3+(2*n+2)*x)*P(n-1,x) + (6*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[((3*n + 3 + (2*n + 2)*x)*P[n - 1, x] + (6*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] = ((3*n + 3 + (2*n + 2)*x)*v[i - 1] + (6*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