Real-rooted atlas

A062264 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(16\,n + 12\,n^{2} + 16\,n^{2} x + 2\,n^{3} + 8\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(48 + 96\,x + 48\,x^{2} + 80\,n + 176\,n x + 100\,n x^{2} + 44\,n^{2} + 132\,n^{2} x + 73\,n^{2} x^{2} + 8\,n^{3} + 42\,n^{3} x + 26\,n^{3} x^{2} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 32\,x^{2} - 16\,x^{3} - 16\,n x - 44\,n x^{2} - 28\,n x^{3} - 4\,n^{2} x - 19\,n^{2} x^{2} - 15\,n^{2} x^{3} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161016
2288445157
31205045041401268
44952640378020163509281
52002128702376018480630075664168
680086006013513513728069300166321470427885
7318242722727207209009006006002162163880826402783980
81259701209312367567254454404504500216216058858882368445517798465
9497420529074018139680310390083063060018378360672672014414401621627150112313280
10196125622881320872972101693036801939938001396755366432426018670080324324030030011011701661693
117726160980628004118637608923714801163962800977728752543182640199536480472586406806800528528163804349045220
1230421755417212640191222460045762640006692786100642507465641825063281862340480561196350110853600134774648910722366026785272705
131197598501764461790876146544022946695200371821450004015671660029983681728157057380485761615860145495350024387792025395552144799233320164107987800
1447143560074251107003970039027511292555456020078358300024094029960020301451170012238237440053006865912163601438003520987470507911040458530802279088459001001087346125
Technical details
Node id
n-6f62feec2f26ccab · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-08a01dd7e1a4e613

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:      6,      10;
  2:     28,      84,       45;
  3:    120,     504,      504,      140;
  4:    495,    2640,     3780,     2016,      350;
  5:   2002,   12870,    23760,    18480,     6300,      756;
  6:   8008,   60060,   135135,   137280,    69300,    16632,    1470;
  7:  31824,  272272,   720720,   900900,   600600,   216216,   38808,    2640;
  8: 125970, 1209312,  3675672,  5445440,  4504500,  2162160,  588588,   82368,   4455;
  9: 497420, 5290740, 18139680, 31039008, 30630600, 18378360, 6726720, 1441440, 162162, 7150;
%F Formula
The row polynomials satisfy (2*n^3+12*n^2+16*n+(n^4+8*n^3+16*n^2)*x)*P(n,x) = (8*n^3+44*n^2+80*n+48+(4*n^4+42*n^3+132*n^2+176*n+96)*x+(3*n^4+26*n^3+73*n^2+100*n+48)*x^2)*P(n-1,x) - ((4*n^2+16*n+16)*x+(2*n^3+19*n^2+44*n+32)*x^2+(2*n^3+15*n^2+28*n+16)*x^3)*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[((8*n^3 + 44*n^2 + 80*n + 48 + (4*n^4 + 42*n^3 + 132*n^2 + 176*n + 96)*x + (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*P[n - 1, x] - ((4*n^2 + 16*n + 16)*x + (2*n^3 + 19*n^2 + 44*n + 32)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 12*n^2 + 16*n + (n^4 + 8*n^3 + 16*n^2)*x)]];
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] = ((8*n^3 + 44*n^2 + 80*n + 48 + (4*n^4 + 42*n^3 + 132*n^2 + 176*n + 96)*x + (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*v[i - 1] - ((4*n^2 + 16*n + 16)*x + (2*n^3 + 19*n^2 + 44*n + 32)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 12*n^2 + 16*n + (n^4 + 8*n^3 + 16*n^2)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.