Real-rooted atlas

A110441 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 2\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-4 - 10\,x - 8\,x^{2} - 4\,n - 12\,n x - 8\,n x^{2} + 4\,n x^{3} - 2\,n^{2} x + 4\,n^{2} x^{2} + 8\,n^{2} x^{3} + 4\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 6\,x - 10\,x^{2} - 4\,x^{3} + 2\,n x^{2} + 8\,n x^{3} + 4\,n x^{4} + 2\,n^{2} x^{2} + 8\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2146448126
3483525762561232
4164166443204096128011524
556072642611238912256006144104592
619123020813939228672029440014745628672928760
765281216006865921812480256000019906568028161310728111744
822288478208319718410338304187136001995571212443648419430458982469933072
9760961847936142848005477990412142592016566681614129561673400320212336642621440596632512
1025980870440966183244827462860872217600012056002561313406976931135488414056448104857600115343365046532064
1188704026558976261024768131872768040189952007957512192105971712009581887488580386816022544384005075107845033164842378913536
1230285449924403210797880326117785600212252672004871002521676896927744849094901766550585344034603008000119265034242415919104218103808353710944320
131034009636808652843925667842759255654410745200640028080635904051370681958467200299827263250838323242504028160019894422732861605937152113413980169395240962936711484672
1435303296135657881617619210240121561677824525320273920154157501644832114630983684866567045120542303531827244421034803202642101141504110971217510431188844544052613349376402653184024270978645888
Technical details
Node id
n-813b80677ca98f88 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-435d714fa3749f4c

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:     4,       8;
  2:    14,      64,       48;
  3:    48,     352,      576,      256;
  4:   164,    1664,     4320,     4096,      1280;
  5:   560,    7264,    26112,    38912,     25600,      6144;
  6:  1912,   30208,   139392,   286720,    294400,    147456,     28672;
  7:  6528,  121600,   686592,  1812480,   2560000,   1990656,    802816,   131072;
  8: 22288,  478208,  3197184, 10338304,  18713600,  19955712,  12443648,  4194304,   589824;
  9: 76096, 1847936, 14284800, 54779904, 121425920, 165666816, 141295616, 73400320, 21233664, 2621440;
%F Formula
The row polynomials satisfy -(2*n+2*n*x-2*n^3*x^2)*P(n,x) = -(4*n+4+(2*n^2+12*n+10)*x-(4*n^3+4*n^2-8*n-8)*x^2-(4*n^3+8*n^2+4*n)*x^3)*P(n-1,x) - (1+6*x-(2*n^2+2*n-10)*x^2-(8*n^2+8*n-4)*x^3-(4*n^2+4*n)*x^4)*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[(-(4*n + 4 + (2*n^2 + 12*n + 10)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*P[n - 1, x] - (1 + 6*x - (2*n^2 + 2*n - 10)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (4*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(-2*n - 2*n*x + 2*n^3*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] = (-(4*n + 4 + (2*n^2 + 12*n + 10)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*v[i - 1] - (1 + 6*x - (2*n^2 + 2*n - 10)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (4*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(-2*n - 2*n*x + 2*n^3*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. [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