Real-rooted atlas

A089627 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-36 + 27\,x - 25\,x^{2} - 2\,x^{3} + 36\,n - 87\,n x + 95\,n x^{2} + 8\,n x^{3} + 60\,n^{2} x - 116\,n^{2} x^{2} + 12\,n^{2} x^{3} - n^{2} x^{4} + 46\,n^{3} x^{2} - 30\,n^{3} x^{3} + 3\,n^{3} x^{4} + 12\,n^{4} x^{3} - 3\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-324 - 153\,x + 243\,x^{2} + 118\,x^{3} + 32\,x^{4} + 180\,n - 729\,n x - 835\,n x^{2} + 52\,n x^{3} + 24\,n x^{4} + 426\,n^{2} x - 230\,n^{2} x^{2} - 672\,n^{2} x^{3} - 85\,n^{2} x^{4} - 8\,n^{2} x^{5} + 302\,n^{3} x^{2} + 166\,n^{3} x^{3} - 109\,n^{3} x^{4} - 12\,n^{3} x^{5} + 72\,n^{4} x^{3} + 61\,n^{4} x^{4} + 5\,n^{5} x^{4} + 4\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-54 - 54\,x + 126\,x^{2} + 178\,x^{3} + 68\,x^{4} + 16\,x^{5} - 216\,n x - 462\,n x^{2} - 178\,n x^{3} + 84\,n x^{4} + 16\,n x^{5} - 222\,n^{2} x^{2} - 554\,n^{2} x^{3} - 386\,n^{2} x^{4} - 62\,n^{2} x^{5} - 8\,n^{2} x^{6} - 72\,n^{3} x^{3} - 180\,n^{3} x^{4} - 124\,n^{3} x^{5} - 16\,n^{3} x^{6} - 6\,n^{4} x^{4} - 14\,n^{4} x^{5} - 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
33811
472431
5199654169
651320270641
7141108013503202891
839335285670224011831
91107114242268013440175050401
10313936576861846720015750208849
1189531162803175203091201102509072871195
122565336740011404801330560635250997923599135
137378911552644018410547008032917507983364527614852905
142129413617952139382102169024015765750518918458858861002865
1561622711292008477296828360352071321250298378085297292219648249917435
Technical details
Node id
n-6b5aa313c13a9a3e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-607fceb0be5e82a1

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    1;
   3:    3,     8;
   4:    7,    24;
   5:   19,    96,    54;
   6:   51,   320,   270;
   7:  141,  1080,  1350,   320;
   8:  393,  3528,  5670,  2240;
   9: 1107, 11424, 22680, 13440,  1750;
  10: 3139, 36576, 86184, 67200, 15750;
%F Formula
The row polynomials satisfy (36*n-36+(60*n^2-87*n+27)*x+(46*n^3-116*n^2+95*n-25)*x^2+(12*n^4-30*n^3+12*n^2+8*n-2)*x^3+(n^5-3*n^4+3*n^3-n^2)*x^4)*P(n,x) = (180*n-324+(426*n^2-729*n-153)*x+(302*n^3-230*n^2-835*n+243)*x^2+(72*n^4+166*n^3-672*n^2+52*n+118)*x^3+(5*n^5+61*n^4-109*n^3-85*n^2+24*n+32)*x^4+(4*n^5-12*n^3-8*n^2)*x^5)*P(n-1,x) - (54+(216*n+54)*x+(222*n^2+462*n-126)*x^2+(72*n^3+554*n^2+178*n-178)*x^3+(6*n^4+180*n^3+386*n^2-84*n-68)*x^4+(14*n^4+124*n^3+62*n^2-16*n-16)*x^5+(8*n^4+16*n^3+8*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((180*n - 324 + (426*n^2 - 729*n - 153)*x + (302*n^3 - 230*n^2 - 835*n + 243)*x^2 + (72*n^4 + 166*n^3 - 672*n^2 + 52*n + 118)*x^3 + (5*n^5 + 61*n^4 - 109*n^3 - 85*n^2 + 24*n + 32)*x^4 + (4*n^5 - 12*n^3 - 8*n^2)*x^5)*P[n - 1, x] - (54 + (216*n + 54)*x + (222*n^2 + 462*n - 126)*x^2 + (72*n^3 + 554*n^2 + 178*n - 178)*x^3 + (6*n^4 + 180*n^3 + 386*n^2 - 84*n - 68)*x^4 + (14*n^4 + 124*n^3 + 62*n^2 - 16*n - 16)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*D[P[n - 1, x], x])/(36*n - 36 + (60*n^2 - 87*n + 27)*x + (46*n^3 - 116*n^2 + 95*n - 25)*x^2 + (12*n^4 - 30*n^3 + 12*n^2 + 8*n - 2)*x^3 + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((180*n - 324 + (426*n^2 - 729*n - 153)*x + (302*n^3 - 230*n^2 - 835*n + 243)*x^2 + (72*n^4 + 166*n^3 - 672*n^2 + 52*n + 118)*x^3 + (5*n^5 + 61*n^4 - 109*n^3 - 85*n^2 + 24*n + 32)*x^4 + (4*n^5 - 12*n^3 - 8*n^2)*x^5)*v[i - 1] - (54 + (216*n + 54)*x + (222*n^2 + 462*n - 126)*x^2 + (72*n^3 + 554*n^2 + 178*n - 178)*x^3 + (6*n^4 + 180*n^3 + 386*n^2 - 84*n - 68)*x^4 + (14*n^4 + 124*n^3 + 62*n^2 - 16*n - 16)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*deriv(v[i - 1], x))/(36*n - 36 + (60*n^2 - 87*n + 27)*x + (46*n^3 - 116*n^2 + 95*n - 25)*x^2 + (12*n^4 - 30*n^3 + 12*n^2 + 8*n - 2)*x^3 + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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