Real-rooted atlas

A207607 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(100 + 400\,x + 432\,x^{2} - 40\,n - 304\,n x - 328\,n x^{2} + 128\,n x^{3} - 120\,n^{2} x^{2} - 192\,n^{2} x^{3} + 16\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n}(x) = \left(240 + 1224\,x + 1920\,x^{2} + 896\,x^{3} + 128\,x^{4} - 60\,n - 436\,n x - 880\,n x^{2} - 544\,n x^{3} - 64\,n x^{4} - 20\,n^{2} x - 296\,n^{2} x^{2} - 464\,n^{2} x^{3} - 128\,n^{2} x^{4} + 24\,n^{3} x^{2} + 112\,n^{3} x^{3} + 64\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-25 - 340\,x - 1332\,x^{2} - 1920\,x^{3} - 896\,x^{4} - 128\,x^{5} - 80\,n x^{2} - 272\,n x^{3} - 256\,n x^{4} - 64\,n x^{5} + 20\,n^{2} x^{2} + 128\,n^{2} x^{3} + 208\,n^{2} x^{4} + 64\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
28362468
32113218064397
4554449126721602243
5144142038884384216038412380
63774392150122284817280633689667141
7987132645432410419210680060096174722048359183

All 15 rows

Technical details
Node id
n-b0a3a11af8a89ff9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c3122552fd8b34ce

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,      8;
  2:    8,     36,     24;
  3:   21,    132,    180,      64;
  4:   55,    444,    912,     672,     160;
  5:  144,   1420,   3888,    4384,    2160,     384;
  6:  377,   4392,  15012,   22848,   17280,    6336,     896;
  7:  987,  13264,  54324,  104192,  106800,   60096,   17472,   2048;
  8: 2584,  39348, 187752,  434592,  563600,  430272,  191744,  46080,   4608;
  9: 6765, 115116, 626976, 1700256, 2670480, 2583360, 1561728, 574464, 117504, 10240;
%F Formula
The row polynomials satisfy -(40*n-100+(304*n-400)*x-(16*n^3-120*n^2-328*n+432)*x^2-(64*n^3-192*n^2+128*n)*x^3)*P(n,x) = -(60*n-240+(20*n^2+436*n-1224)*x-(24*n^3-296*n^2-880*n+1920)*x^2-(112*n^3-464*n^2-544*n+896)*x^3-(64*n^3-128*n^2-64*n+128)*x^4)*P(n-1,x) - (25+340*x-(20*n^2-80*n-1332)*x^2-(128*n^2-272*n-1920)*x^3-(208*n^2-256*n-896)*x^4-(64*n^2-64*n-128)*x^5)*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[(-(60*n - 240 + (20*n^2 + 436*n - 1224)*x - (24*n^3 - 296*n^2 - 880*n + 1920)*x^2 - (112*n^3 - 464*n^2 - 544*n + 896)*x^3 - (64*n^3 - 128*n^2 - 64*n + 128)*x^4)*P[n - 1, x] - (25 + 340*x - (20*n^2 - 80*n - 1332)*x^2 - (128*n^2 - 272*n - 1920)*x^3 - (208*n^2 - 256*n - 896)*x^4 - (64*n^2 - 64*n - 128)*x^5)*D[P[n - 1, x], x])/(-40*n + 100 - (304*n - 400)*x + (16*n^3 - 120*n^2 - 328*n + 432)*x^2 + (64*n^3 - 192*n^2 + 128*n)*x^3)]];
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] = (-(60*n - 240 + (20*n^2 + 436*n - 1224)*x - (24*n^3 - 296*n^2 - 880*n + 1920)*x^2 - (112*n^3 - 464*n^2 - 544*n + 896)*x^3 - (64*n^3 - 128*n^2 - 64*n + 128)*x^4)*v[i - 1] - (25 + 340*x - (20*n^2 - 80*n - 1332)*x^2 - (128*n^2 - 272*n - 1920)*x^3 - (208*n^2 - 256*n - 896)*x^4 - (64*n^2 - 64*n - 128)*x^5)*deriv(v[i - 1], x))/(-40*n + 100 - (304*n - 400)*x + (16*n^3 - 120*n^2 - 328*n + 432)*x^2 + (64*n^3 - 192*n^2 + 128*n)*x^3)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023