Real-rooted atlas

A232535 → reverse → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n^{2} + 4\,n^{2} x + 6\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n}(x) = \left(-3\,x - 18\,x^{2} - 42\,x^{3} - 48\,x^{4} - 27\,x^{5} - 6\,x^{6} + n + 8\,n x + 34\,n x^{2} + 76\,n x^{3} + 89\,n x^{4} + 52\,n x^{5} + 12\,n x^{6} + n^{2} + 6\,n^{2} x + 8\,n^{2} x^{2} - 8\,n^{2} x^{3} - 27\,n^{2} x^{4} - 22\,n^{2} x^{5} - 6\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 18\,x^{3} + 42\,x^{4} + 48\,x^{5} + 27\,x^{6} + 6\,x^{7} + 2\,n x + 10\,n x^{2} + 20\,n x^{3} + 20\,n x^{4} + 10\,n x^{5} + 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(6\,x^{3} + 30\,x^{4} + 60\,x^{5} + 60\,x^{6} + 30\,x^{7} + 6\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
23141229
34378048169
4576296416192985
56135816196820487685741
6721818766784117769728307233461
7832938081910448640657924505612288195025
89472705646656161920316416350208204800491521136689
910651121921024324618241214720192102417981449175041966086625109
101187019932206976117145639603208350720110755848978432406323278643238613965
111211333115239124827089921139174430621696537559046134169643843584178257923145728225058681
12131444469047001285811520296591369845964821892300832872857632899072021023948877594624125829121311738121
131418076843211966241171456071172608284934144777928704147180748819290521601718616064993001472335544320503316487645370045
14152226971881966848224040961595443207560990722476670976572660121694163435521094320128087828725764630511616144284057620132659244560482149
Technical details
Node id
n-868665bd75540376 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2ab9eb92c097ff31

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:  2,   3;
  2:  3,  14,    12;
  3:  4,  37,    80,     48;
  4:  5,  76,   296,    416,    192;
  5:  6, 135,   816,   1968,   2048,     768;
  6:  7, 218,  1876,   6784,  11776,    9728,    3072;
  7:  8, 329,  3808,  19104,  48640,   65792,   45056,   12288;
  8:  9, 472,  7056,  46656, 161920,  316416,  350208,  204800,  49152;
  9: 10, 651, 12192, 102432, 461824, 1214720, 1921024, 1798144, 917504, 196608;
%F Formula
The row polynomials satisfy (n^2+4*n^2*x+6*n^2*x^2+4*n^2*x^3+n^2*x^4)*P(n,x) = (n^2+n+(6*n^2+8*n-3)*x+(8*n^2+34*n-18)*x^2-(8*n^2-76*n+42)*x^3-(27*n^2-89*n+48)*x^4-(22*n^2-52*n+27)*x^5-(6*n^2-12*n+6)*x^6)*P(n-1,x) + (2*n*x+(10*n+3)*x^2+(20*n+18)*x^3+(20*n+42)*x^4+(10*n+48)*x^5+(2*n+27)*x^6+6*x^7)*d/dx P(n-1,x) + (6*x^3+30*x^4+60*x^5+60*x^6+30*x^7+6*x^8)*d^2/dx^2 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[((n^2 + n + (6*n^2 + 8*n - 3)*x + (8*n^2 + 34*n - 18)*x^2 - (8*n^2 - 76*n + 42)*x^3 - (27*n^2 - 89*n + 48)*x^4 - (22*n^2 - 52*n + 27)*x^5 - (6*n^2 - 12*n + 6)*x^6)*P[n - 1, x] + (2*n*x + (10*n + 3)*x^2 + (20*n + 18)*x^3 + (20*n + 42)*x^4 + (10*n + 48)*x^5 + (2*n + 27)*x^6 + 6*x^7)*D[P[n - 1, x], x] + (6*x^3 + 30*x^4 + 60*x^5 + 60*x^6 + 30*x^7 + 6*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 4*n^2*x + 6*n^2*x^2 + 4*n^2*x^3 + n^2*x^4)]];
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] = ((n^2 + n + (6*n^2 + 8*n - 3)*x + (8*n^2 + 34*n - 18)*x^2 - (8*n^2 - 76*n + 42)*x^3 - (27*n^2 - 89*n + 48)*x^4 - (22*n^2 - 52*n + 27)*x^5 - (6*n^2 - 12*n + 6)*x^6)*v[i - 1] + (2*n*x + (10*n + 3)*x^2 + (20*n + 18)*x^3 + (20*n + 42)*x^4 + (10*n + 48)*x^5 + (2*n + 27)*x^6 + 6*x^7)*deriv(v[i - 1], x) + (6*x^3 + 30*x^4 + 60*x^5 + 60*x^6 + 30*x^7 + 6*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 4*n^2*x + 6*n^2*x^2 + 4*n^2*x^3 + n^2*x^4)); 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

Candidate equalities ?

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