Real-rooted atlas

A207606 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-360\,x - 528\,x^{2} + 192\,x^{3} - 40\,n - 264\,n x - 384\,n x^{2} + 416\,n x^{3} + 288\,n^{2} x^{3} + 16\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n}(x) = \left(40 - 536\,x - 2128\,x^{2} - 1024\,x^{3} + 384\,x^{4} - 60\,n - 476\,n x - 1464\,n x^{2} - 32\,n x^{3} + 704\,n x^{4} - 20\,n^{2} x - 96\,n^{2} x^{2} + 464\,n^{2} x^{3} + 384\,n^{2} x^{4} + 24\,n^{3} x^{2} + 112\,n^{3} x^{3} + 64\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-25 - 290\,x - 972\,x^{2} - 904\,x^{3} + 256\,x^{4} + 192\,x^{5} + 20\,n x^{2} + 368\,n x^{3} + 784\,n x^{4} + 256\,n x^{5} + 20\,n^{2} x^{2} + 128\,n^{2} x^{3} + 208\,n^{2} x^{4} + 64\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 44\,x + 24\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
013442481
13417620464478
28962011167361602721
3233204050045120232038415101
46106432200162796819600672089682242
51597196967434013216012640066816183682048441425
64181590442620925667526900004970882101124812846082342005
71094617416088906822670083360480308044817718406225921221121024012308894

All 15 rows

Technical details
Node id
n-7d6e69863e6bcaa2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-78c93daabb7301a9

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:    13,      44,      24;
  1:    34,     176,     204,       64;
  2:    89,     620,    1116,      736,      160;
  3:   233,    2040,    5004,     5120,     2320,      384;
  4:   610,    6432,   20016,    27968,    19600,     6720,      896;
  5:  1597,   19696,   74340,   132160,   126400,    66816,    18368,     2048;
  6:  4181,   59044,  262092,   566752,   690000,   497088,   210112,    48128,     4608;
  7: 10946,  174160,  889068,  2267008,  3360480,  3080448,  1771840,   622592,   122112,   10240;
  8: 28657,  507220, 2927592,  8605120, 15058960, 16809408, 12293120,  5869568,  1762560,  302080,  22528;
  9: 75025, 1462128, 9415368, 31355264, 63329200, 83489280, 74329472, 45084672, 18372096, 4812800, 732160, 49152;
%F Formula
The row polynomials satisfy -(40*n+(264*n+360)*x-(16*n^3-384*n-528)*x^2-(64*n^3+288*n^2+416*n+192)*x^3)*P(n,x) = -(60*n-40+(20*n^2+476*n+536)*x-(24*n^3-96*n^2-1464*n-2128)*x^2-(112*n^3+464*n^2-32*n-1024)*x^3-(64*n^3+384*n^2+704*n+384)*x^4)*P(n-1,x) - (25+290*x-(20*n^2+20*n-972)*x^2-(128*n^2+368*n-904)*x^3-(208*n^2+784*n+256)*x^4-(64*n^2+256*n+192)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 44*x + 24*x^2.
%t Mathematica
P[0, x_] := 13 + 44*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(60*n - 40 + (20*n^2 + 476*n + 536)*x - (24*n^3 - 96*n^2 - 1464*n - 2128)*x^2 - (112*n^3 + 464*n^2 - 32*n - 1024)*x^3 - (64*n^3 + 384*n^2 + 704*n + 384)*x^4)*P[n - 1, x] - (25 + 290*x - (20*n^2 + 20*n - 972)*x^2 - (128*n^2 + 368*n - 904)*x^3 - (208*n^2 + 784*n + 256)*x^4 - (64*n^2 + 256*n + 192)*x^5)*D[P[n - 1, x], x])/(-40*n - (264*n + 360)*x + (16*n^3 - 384*n - 528)*x^2 + (64*n^3 + 288*n^2 + 416*n + 192)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 44*x + 24*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(60*n - 40 + (20*n^2 + 476*n + 536)*x - (24*n^3 - 96*n^2 - 1464*n - 2128)*x^2 - (112*n^3 + 464*n^2 - 32*n - 1024)*x^3 - (64*n^3 + 384*n^2 + 704*n + 384)*x^4)*v[i - 1] - (25 + 290*x - (20*n^2 + 20*n - 972)*x^2 - (128*n^2 + 368*n - 904)*x^3 - (208*n^2 + 784*n + 256)*x^4 - (64*n^2 + 256*n + 192)*x^5)*deriv(v[i - 1], x))/(-40*n - (264*n + 360)*x + (16*n^3 - 384*n - 528)*x^2 + (64*n^3 + 288*n^2 + 416*n + 192)*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