Real-rooted atlas

A207606 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(54 + 72\,n + 24\,n^{2}\right)\,P_{n}(x) = \left(162 + 207\,x + 57\,x^{2} + 216\,n + 300\,n x + 95\,n x^{2} + 5\,n x^{3} + 72\,n^{2} + 108\,n^{2} x + 38\,n^{2} x^{2} + 5\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-54\,x - 135\,x^{2} - 57\,x^{3} - 36\,n x - 90\,n x^{2} - 56\,n x^{3} - 10\,n x^{4}\right)\,P_{n-1}'(x) + \left(9\,x^{3} + 18\,x^{4} + 5\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 18 + 33\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
018331364
15415313234373
2162621837465892174
3486234943203753153023312671
414588505198452359815012482461073852
54374298898456412798011151055755147721597430441
6131221027893419016291276986254781971965694428341812508794
739366347733132969628831953898692340248619127886668011306201094614622323

All 15 rows

Technical details
Node id
n-64ab05af64bea805 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-55426929a3969ca5

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:     18,      33,       13;
  1:     54,     153,      132,       34;
  2:    162,     621,      837,      465,       89;
  3:    486,    2349,     4320,     3753,     1530,       233;
  4:   1458,    8505,    19845,    23598,    15012,      4824,       610;
  5:   4374,   29889,    84564,   127980,   111510,     55755,     14772,     1597;
  6:  13122,  102789,   341901,   629127,   698625,    478197,    196569,    44283,     4181;
  7:  39366,  347733,  1329696,  2883195,  3898692,   3402486,   1912788,   666801,   130620,   10946;
  8: 118098, 1161297,  5019165, 12535884, 20003760,  21274407,  15247197,  7260570,  2195694,  380415,   28657;
  9: 354294, 3838185, 18502020, 52317414, 96289236, 120951306, 105666120, 64120815, 26456004, 7061526, 1096596, 75025;
%F Formula
The row polynomials satisfy (24*n^2+72*n+54)*P(n,x) = (72*n^2+216*n+162+(108*n^2+300*n+207)*x+(38*n^2+95*n+57)*x^2+(5*n^2+5*n)*x^3)*P(n-1,x) - ((36*n+54)*x+(90*n+135)*x^2+(56*n+57)*x^3+10*n*x^4)*d/dx P(n-1,x) + (9*x^3+18*x^4+5*x^5)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 18 + 33*x + 13*x^2.
%t Mathematica
P[0, x_] := 18 + 33*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72*n^2 + 216*n + 162 + (108*n^2 + 300*n + 207)*x + (38*n^2 + 95*n + 57)*x^2 + (5*n^2 + 5*n)*x^3)*P[n - 1, x] - ((36*n + 54)*x + (90*n + 135)*x^2 + (56*n + 57)*x^3 + 10*n*x^4)*D[P[n - 1, x], x] + (9*x^3 + 18*x^4 + 5*x^5)*D[P[n - 1, x], {x, 2}])/(24*n^2 + 72*n + 54)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 18 + 33*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72*n^2 + 216*n + 162 + (108*n^2 + 300*n + 207)*x + (38*n^2 + 95*n + 57)*x^2 + (5*n^2 + 5*n)*x^3)*v[i - 1] - ((36*n + 54)*x + (90*n + 135)*x^2 + (56*n + 57)*x^3 + 10*n*x^4)*deriv(v[i - 1], x) + (9*x^3 + 18*x^4 + 5*x^5)*deriv(deriv(v[i - 1], x), x))/(24*n^2 + 72*n + 54)); 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