Real-rooted atlas

A111125 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-333 + 81\,x^{2} + 465\,n - 54\,n x^{2} - 243\,n^{2} + 54\,n^{3}\right)\,P_{n}(x) = \left(-69 - 9\,x + 135\,x^{2} + 81\,x^{3} + 180\,n + 90\,n x - 162\,n x^{2} - 81\,n x^{3} - 135\,n^{2} - 162\,n^{2} x + 81\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(50 + 90\,x - 279\,x^{2} - 324\,x^{3} - 81\,x^{4} - 90\,n - 324\,n x - 162\,n x^{2} + 45\,n^{2} + 162\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
27310
33239980
412227317127593
542114251593675814195
613696300105037830251124328756
74283250925651162262348308991729192698
813043931112663013943353240001450713134721871269395
938938328407114387321452312359530154475157663910716365618251093
1011448411150254590099104703301454986812711573691456522132443608551968353059726
1133256136755251748633447121210795325238760320163430290295070048266860120066359049338215220
1295662711833488639466381991549163968015855301370624830805982985123781212866463021340539562831771472140056773
13272955737372464226265454800659620184308412529043442623206102634248946647413412980354837097251084730131293173153144113456968535
14773532711615796377918311830904804268082709335147083426371913439979817978504688121557680915804713530188171448338370040241983839159432384166987960
15217940363561957752623050549115330832103381315742869891102801104939247465116104400784948965653355680131949824351866649716913909013402942021355174554782969523981517246
Technical details
Node id
n-856f40b42f6b9edb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-622256dab047377a

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      7,       3;
   3:     32,      39,       9;
   4:    122,     273,     171,       27;
   5:    421,    1425,    1593,      675,       81;
   6:   1369,    6300,   10503,     7830,     2511,      243;
   7:   4283,   25092,   56511,    62262,    34830,     8991,     729;
   8:  13043,   93111,  266301,   394335,   324000,   145071,   31347,    2187;
   9:  38938,  328407, 1143873,  2145231,  2359530,  1544751,  576639,  107163,   6561;
  10: 114484, 1115025, 4590099, 10470330, 14549868, 12711573, 6914565, 2213244, 360855, 19683;
%F Formula
The row polynomials satisfy (54*n^3-243*n^2+465*n-333-(54*n-81)*x^2)*P(n,x) = (81*n^3-135*n^2+180*n-69+(81*n^3-162*n^2+90*n-9)*x-(162*n-135)*x^2-(81*n-81)*x^3)*P(n-1,x) + (45*n^2-90*n+50+(162*n^2-324*n+90)*x+(81*n^2-162*n-279)*x^2-324*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 - 135*n^2 + 180*n - 69 + (81*n^3 - 162*n^2 + 90*n - 9)*x - (162*n - 135)*x^2 - (81*n - 81)*x^3)*P[n - 1, x] + (45*n^2 - 90*n + 50 + (162*n^2 - 324*n + 90)*x + (81*n^2 - 162*n - 279)*x^2 - 324*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 - 243*n^2 + 465*n - 333 - (54*n - 81)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n^3 - 135*n^2 + 180*n - 69 + (81*n^3 - 162*n^2 + 90*n - 9)*x - (162*n - 135)*x^2 - (81*n - 81)*x^3)*v[i - 1] + (45*n^2 - 90*n + 50 + (162*n^2 - 324*n + 90)*x + (81*n^2 - 162*n - 279)*x^2 - 324*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 - 243*n^2 + 465*n - 333 - (54*n - 81)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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