Real-rooted atlas

A391403 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36\,n + 33\,n x + n x^{2} - 10\,n x^{3} + 60\,n^{2} x + 22\,n^{2} x^{2} - 6\,n^{2} x^{3} + 46\,n^{3} x^{2} + 18\,n^{3} x^{3} + n^{3} x^{4} + 12\,n^{4} x^{3} + 2\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-144 - 456\,x - 520\,x^{2} - 264\,x^{3} - 72\,x^{4} - 16\,x^{5} + 180\,n + 123\,n x - 389\,n x^{2} - 506\,n x^{3} - 204\,n x^{4} - 32\,n x^{5} + 426\,n^{2} x + 676\,n^{2} x^{2} + 258\,n^{2} x^{3} + 4\,n^{2} x^{4} - 4\,n^{2} x^{5} + 302\,n^{3} x^{2} + 454\,n^{3} x^{3} + 185\,n^{3} x^{4} + 28\,n^{3} x^{5} + 72\,n^{4} x^{3} + 86\,n^{4} x^{4} + 20\,n^{4} x^{5} + 5\,n^{5} x^{4} + 4\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-54 - 270\,x - 558\,x^{2} - 626\,x^{3} - 420\,x^{4} - 168\,x^{5} - 32\,x^{6} - 216\,n x - 906\,n x^{2} - 1502\,n x^{3} - 1252\,n x^{4} - 536\,n x^{5} - 96\,n x^{6} - 222\,n^{2} x^{2} - 770\,n^{2} x^{3} - 962\,n^{2} x^{4} - 518\,n^{2} x^{5} - 104\,n^{2} x^{6} - 72\,n^{3} x^{3} - 204\,n^{3} x^{4} - 180\,n^{3} x^{5} - 48\,n^{3} x^{6} - 6\,n^{4} x^{4} - 14\,n^{4} x^{5} - 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
23811
372431
4199654169
551320270641
6141108013503202891
739335285670224011831
81107114242268013440175050401
9313936576861846720015750208849
1089531162803175203091201102509072871195
112565336740011404801330560635250997923599135
127378911552644018410547008032917507983364527614852905
132129413617952139382102169024015765750518918458858861002865
1461622711292008477296828360352071321250298378085297292219648249917435
Technical details
Node id
n-2652387edb474022 · 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 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:    1;
  2:    3,     8;
  3:    7,    24;
  4:   19,    96,    54;
  5:   51,   320,   270;
  6:  141,  1080,  1350,   320;
  7:  393,  3528,  5670,  2240;
  8: 1107, 11424, 22680, 13440,  1750;
  9: 3139, 36576, 86184, 67200, 15750;
%F Formula
The row polynomials satisfy (36*n+(60*n^2+33*n)*x+(46*n^3+22*n^2+n)*x^2+(12*n^4+18*n^3-6*n^2-10*n)*x^3+(n^5+2*n^4+n^3)*x^4)*P(n,x) = (180*n-144+(426*n^2+123*n-456)*x+(302*n^3+676*n^2-389*n-520)*x^2+(72*n^4+454*n^3+258*n^2-506*n-264)*x^3+(5*n^5+86*n^4+185*n^3+4*n^2-204*n-72)*x^4+(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16)*x^5)*P(n-1,x) - (54+(216*n+270)*x+(222*n^2+906*n+558)*x^2+(72*n^3+770*n^2+1502*n+626)*x^3+(6*n^4+204*n^3+962*n^2+1252*n+420)*x^4+(14*n^4+180*n^3+518*n^2+536*n+168)*x^5+(8*n^4+48*n^3+104*n^2+96*n+32)*x^6)*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[((180*n - 144 + (426*n^2 + 123*n - 456)*x + (302*n^3 + 676*n^2 - 389*n - 520)*x^2 + (72*n^4 + 454*n^3 + 258*n^2 - 506*n - 264)*x^3 + (5*n^5 + 86*n^4 + 185*n^3 + 4*n^2 - 204*n - 72)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*P[n - 1, x] - (54 + (216*n + 270)*x + (222*n^2 + 906*n + 558)*x^2 + (72*n^3 + 770*n^2 + 1502*n + 626)*x^3 + (6*n^4 + 204*n^3 + 962*n^2 + 1252*n + 420)*x^4 + (14*n^4 + 180*n^3 + 518*n^2 + 536*n + 168)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*D[P[n - 1, x], x])/(36*n + (60*n^2 + 33*n)*x + (46*n^3 + 22*n^2 + n)*x^2 + (12*n^4 + 18*n^3 - 6*n^2 - 10*n)*x^3 + (n^5 + 2*n^4 + n^3)*x^4)]];
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] = ((180*n - 144 + (426*n^2 + 123*n - 456)*x + (302*n^3 + 676*n^2 - 389*n - 520)*x^2 + (72*n^4 + 454*n^3 + 258*n^2 - 506*n - 264)*x^3 + (5*n^5 + 86*n^4 + 185*n^3 + 4*n^2 - 204*n - 72)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*v[i - 1] - (54 + (216*n + 270)*x + (222*n^2 + 906*n + 558)*x^2 + (72*n^3 + 770*n^2 + 1502*n + 626)*x^3 + (6*n^4 + 204*n^3 + 962*n^2 + 1252*n + 420)*x^4 + (14*n^4 + 180*n^3 + 518*n^2 + 536*n + 168)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*deriv(v[i - 1], x))/(36*n + (60*n^2 + 33*n)*x + (46*n^3 + 22*n^2 + n)*x^2 + (12*n^4 + 18*n^3 - 6*n^2 - 10*n)*x^3 + (n^5 + 2*n^4 + n^3)*x^4)); 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

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