Real-rooted atlas

A207607 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 - 64\,x + 60\,n + 192\,n x - 60\,n^{2} - 192\,n^{2} x + 8\,n^{3} + 64\,n^{3} x\right)\,P_{n}(x) = \left(-24\,n - 48\,n x - 84\,n^{2} - 256\,n^{2} x - 128\,n^{2} x^{2} + 12\,n^{3} + 112\,n^{3} x + 128\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-15 - 72\,x - 48\,x^{2} - 20\,n - 136\,n x - 256\,n x^{2} - 128\,n x^{3} + 5\,n^{2} + 64\,n^{2} x + 208\,n^{2} x^{2} + 128\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
188
23664100
3132480384996
44442432403220488956
514201036826304276481024075980
643924003213708822118416896049152620808
713264144864625152136704016025609584642293764940720
8393485006722607552721408011473920105185285160960104857638563636
911511616719361020153634182144688896008567193664339968267386884718592296529516
10333060543606438028672149740544366105600577167360587661312373293056134479872209715202253217060
1195490817300736136500864617859072177812992034031370244392091648377959219220785397766606028809227468816956983708
122717136541054084752783362431841280805757952018167267328284699197443104361676823109304320111987916803183476736402653184126596551440
13768207216675257616139665929212482560345538150408977504665616579824844821949946265620694564864013564379136058732838912150994944001744830464938794060376
14216003245076088965367136896338134282241416662784004170161848328878750433281385389621248158547522355213154910208007698477219843011845816327066563379275161927686921837276676
Technical details
Node id
n-e3665ba6bd0a4e9f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4cf1c3a9fbbca61f

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      8;
  2:     36,      64;
  3:    132,     480,      384;
  4:    444,    2432,     4032,     2048;
  5:   1420,   10368,    26304,    27648,    10240;
  6:   4392,   40032,   137088,   221184,   168960,    49152;
  7:  13264,  144864,   625152,  1367040,  1602560,   958464,   229376;
  8:  39348,  500672,  2607552,  7214080, 11473920, 10518528,  5160960,  1048576;
  9: 115116, 1671936, 10201536, 34182144, 68889600, 85671936, 64339968, 26738688, 4718592;
%F Formula
The row polynomials satisfy (8*n^3-60*n^2+60*n-8+(64*n^3-192*n^2+192*n-64)*x)*P(n,x) = (12*n^3-84*n^2-24*n+(112*n^3-256*n^2-48*n)*x+(128*n^3-128*n^2)*x^2)*P(n-1,x) + (5*n^2-20*n-15+(64*n^2-136*n-72)*x+(208*n^2-256*n-48)*x^2+(128*n^2-128*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^3 - 84*n^2 - 24*n + (112*n^3 - 256*n^2 - 48*n)*x + (128*n^3 - 128*n^2)*x^2)*P[n - 1, x] + (5*n^2 - 20*n - 15 + (64*n^2 - 136*n - 72)*x + (208*n^2 - 256*n - 48)*x^2 + (128*n^2 - 128*n)*x^3)*D[P[n - 1, x], x])/(8*n^3 - 60*n^2 + 60*n - 8 + (64*n^3 - 192*n^2 + 192*n - 64)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^3 - 84*n^2 - 24*n + (112*n^3 - 256*n^2 - 48*n)*x + (128*n^3 - 128*n^2)*x^2)*v[i - 1] + (5*n^2 - 20*n - 15 + (64*n^2 - 136*n - 72)*x + (208*n^2 - 256*n - 48)*x^2 + (128*n^2 - 128*n)*x^3)*deriv(v[i - 1], x))/(8*n^3 - 60*n^2 + 60*n - 8 + (64*n^3 - 192*n^2 + 192*n - 64)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023