Real-rooted atlas

A172431 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(6\,n\right)\,P_{n}(x) = \left(18 + 21\,x + 5\,x^{2} + 18\,n - 3\,n x - 5\,n x^{2}\right)\,P_{n-1}(x) + \left(36\,x + 27\,x^{2} + 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
22730865
310819811421441
44051080999390552929
51458526569124266126014419305
65103238144139135748165243924377126881
7174961020602245322535301625405999411910987833049

All 15 rows

Technical details
Node id
n-92ce909d52c434fe · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-53dcb76f02d19ad5

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,       3;
  2:     27,      30,       8;
  3:    108,     198,     114,      21;
  4:    405,    1080,     999,     390,      55;
  5:   1458,    5265,    6912,    4266,    1260,     144;
  6:   5103,   23814,   41391,   35748,   16524,    3924,     377;
  7:  17496,  102060,  224532,  253530,  162540,   59994,   11910,    987;
  8:  59049,  419904, 1132866, 1600884, 1331235,  677376,  207981,  35466,   2584;
  9: 196830, 1673055, 5406264, 9272880, 9581490, 6289245, 2650320, 696402, 104070, 6765;
%F Formula
The row polynomials satisfy 6*n*P(n,x) = (18*n+18-(3*n-21)*x-(5*n-5)*x^2)*P(n-1,x) + (36*x+27*x^2+5*x^3)*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[((18*n + 18 - (3*n - 21)*x - (5*n - 5)*x^2)*P[n - 1, x] + (36*x + 27*x^2 + 5*x^3)*D[P[n - 1, x], x])/(6*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] = ((18*n + 18 - (3*n - 21)*x - (5*n - 5)*x^2)*v[i - 1] + (36*x + 27*x^2 + 5*x^3)*deriv(v[i - 1], x))/(6*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