Real-rooted atlas

A232535 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(12 + 67\,x + 145\,x^{2} + 153\,x^{3} + 79\,x^{4} + 16\,x^{5} - 23\,n - 123\,n x - 258\,n x^{2} - 263\,n x^{3} - 129\,n x^{4} - 24\,n x^{5} + 14\,n^{2} + 71\,n^{2} x + 134\,n^{2} x^{2} + 119\,n^{2} x^{3} + 50\,n^{2} x^{4} + 8\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(6 + 59\,x + 208\,x^{2} + 362\,x^{3} + 338\,x^{4} + 163\,x^{5} + 32\,x^{6} - 4\,n - 38\,n x - 134\,n x^{2} - 230\,n x^{3} - 206\,n x^{4} - 92\,n x^{5} - 16\,n x^{6}\right)\,P_{n-1}'(x) + \left(2\,x + 18\,x^{2} + 60\,x^{3} + 100\,x^{4} + 90\,x^{5} + 42\,x^{6} + 8\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13912
2126681159
3483849577292118
41922016756011892656128221
576899844996811971213655159049376032
6307247616297984960384167528414946785314415010459
71228822118416611846731520158552642165414415843585478296966762138

All 15 rows

Technical details
Node id
n-f2ad15cf169afade · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c115e0cff0f472ac

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,       9;
  2:     12,      66,       81;
  3:     48,     384,      957,       729;
  4:    192,    2016,     7560,     11892,      6561;
  5:    768,    9984,    49968,    119712,    136551,      59049;
  6:   3072,   47616,   297984,    960384,   1675284,    1494678,     531441;
  7:  12288,  221184,  1661184,   6731520,  15855264,   21654144,   15843585,    4782969;
  8:  49152, 1007616,  8828928,  43109376, 128054400,  236525760,  264838224,  164115624,   43046721;
  9: 196608, 4521984, 45281280, 258736128, 928360704, 2166063360, 3281913120, 3109957440, 1670750859, 387420489;
%F Formula
The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = (14*n^2-23*n+12+(71*n^2-123*n+67)*x+(134*n^2-258*n+145)*x^2+(119*n^2-263*n+153)*x^3+(50*n^2-129*n+79)*x^4+(8*n^2-24*n+16)*x^5)*P(n-1,x) - (4*n-6+(38*n-59)*x+(134*n-208)*x^2+(230*n-362)*x^3+(206*n-338)*x^4+(92*n-163)*x^5+(16*n-32)*x^6)*d/dx P(n-1,x) + (2*x+18*x^2+60*x^3+100*x^4+90*x^5+42*x^6+8*x^7)*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[((14*n^2 - 23*n + 12 + (71*n^2 - 123*n + 67)*x + (134*n^2 - 258*n + 145)*x^2 + (119*n^2 - 263*n + 153)*x^3 + (50*n^2 - 129*n + 79)*x^4 + (8*n^2 - 24*n + 16)*x^5)*P[n - 1, x] - (4*n - 6 + (38*n - 59)*x + (134*n - 208)*x^2 + (230*n - 362)*x^3 + (206*n - 338)*x^4 + (92*n - 163)*x^5 + (16*n - 32)*x^6)*D[P[n - 1, x], x] + (2*x + 18*x^2 + 60*x^3 + 100*x^4 + 90*x^5 + 42*x^6 + 8*x^7)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n^2*x + n^2*x^2)]];
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] = ((14*n^2 - 23*n + 12 + (71*n^2 - 123*n + 67)*x + (134*n^2 - 258*n + 145)*x^2 + (119*n^2 - 263*n + 153)*x^3 + (50*n^2 - 129*n + 79)*x^4 + (8*n^2 - 24*n + 16)*x^5)*v[i - 1] - (4*n - 6 + (38*n - 59)*x + (134*n - 208)*x^2 + (230*n - 362)*x^3 + (206*n - 338)*x^4 + (92*n - 163)*x^5 + (16*n - 32)*x^6)*deriv(v[i - 1], x) + (2*x + 18*x^2 + 60*x^3 + 100*x^4 + 90*x^5 + 42*x^6 + 8*x^7)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*n^2*x + n^2*x^2)); 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

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