Real-rooted atlas

A060920 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(7 + 13\,x + 5\,x^{2} - 4\,n - 11\,n x - 5\,n x^{2}\right)\,P_{n-1}(x) + \left(18 + 37\,x + 24\,x^{2} + 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21115531
343885913203
4171465468207341345
5683231431042063680898933
627311107518559164628157214223359359
7109235158010366511500576095300426555610394475

All 15 rows

Technical details
Node id
n-370c1859717256b5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bfdae63ac656708a

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:      3,       2;
  2:     11,      15,       5;
  3:     43,      88,      59,      13;
  4:    171,     465,     468,     207,      34;
  5:    683,    2314,    3104,    2063,     680,      89;
  6:   2731,   11075,   18559,   16462,    8157,    2142,     233;
  7:  10923,   51580,  103665,  115005,   76095,   30042,    6555,    610;
  8:  43691,  235445,  551798,  734861,  608455,  320837,  105244,  19641,  1597;
  9: 174763, 1058286, 2833422, 4403439, 4378793, 2889997, 1266248, 355235, 57913, 4181;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = -(4*n-7+(11*n-13)*x+(5*n-5)*x^2)*P(n-1,x) + (18+37*x+24*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[(-(4*n - 7 + (11*n - 13)*x + (5*n - 5)*x^2)*P[n - 1, x] + (18 + 37*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = (-(4*n - 7 + (11*n - 13)*x + (5*n - 5)*x^2)*v[i - 1] + (18 + 37*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*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

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