Real-rooted atlas

A078812 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-200 - 30\,x + 82\,x^{2} + 30\,x^{3} - 2\,x^{4} + 200\,n + 30\,n x - 282\,n x^{2} - 90\,n x^{3} + 10\,n x^{4} + 300\,n^{2} x^{2} + 90\,n^{2} x^{3} - 20\,n^{2} x^{4} - 100\,n^{3} x^{2} - 30\,n^{3} x^{3} + 20\,n^{3} x^{4} - 10\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(300\,n + 95\,n x - 33\,n x^{2} - 19\,n x^{3} - 3\,n x^{4} + 175\,n^{2} x + 285\,n^{2} x^{2} + 125\,n^{2} x^{3} + 15\,n^{2} x^{4} - 150\,n^{3} x^{2} - 90\,n^{3} x^{3} - 9\,n^{3} x^{4} + n^{3} x^{5} - 10\,n^{4} x^{3} - 12\,n^{4} x^{4} - 2\,n^{4} x^{5} + 3\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 225\,x + 110\,x^{2} - 6\,x^{3} - 19\,x^{4} - 3\,x^{5} + 175\,n x^{2} + 285\,n x^{3} + 125\,n x^{4} + 15\,n x^{5} - 175\,n^{2} x^{2} - 285\,n^{2} x^{3} - 120\,n^{2} x^{4} - 9\,n^{2} x^{5} + n^{2} x^{6} - 10\,n^{3} x^{4} - 12\,n^{3} x^{5} - 2\,n^{3} x^{6} + 5\,n^{4} x^{4} + 6\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
3824941
4211008116218
555360459192251091
6144120021061376375365237
737738168559768032506484924379
8987117563215736880213756588102964110836

All 15 rows

Technical details
Node id
n-328963ad68d89b16 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-543cfd9681673a07

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 -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:    3,      4;
   3:    8,     24,      9;
   4:   21,    100,     81,     16;
   5:   55,    360,    459,    192,     25;
   6:  144,   1200,   2106,   1376,    375,     36;
   7:  377,   3816,   8559,   7680,   3250,    648,     49;
   8:  987,  11756,  32157,  36880,  21375,   6588,   1029,    64;
   9: 2584,  35400, 114363, 160128, 118500,  49896,  12005,  1536,   81;
  10: 6765, 104780, 390582, 646816, 584325, 313740, 102900, 20224, 2187, 100;
%F Formula
The row polynomials satisfy (200*n-200+(30*n-30)*x-(100*n^3-300*n^2+282*n-82)*x^2-(30*n^3-90*n^2+90*n-30)*x^3+(2*n^5-10*n^4+20*n^3-20*n^2+10*n-2)*x^4)*P(n,x) = (300*n+(175*n^2+95*n)*x-(150*n^3-285*n^2+33*n)*x^2-(10*n^4+90*n^3-125*n^2+19*n)*x^3+(3*n^5-12*n^4-9*n^3+15*n^2-3*n)*x^4+(n^5-2*n^4+n^3)*x^5)*P(n-1,x) + (125+225*x-(175*n^2-175*n-110)*x^2-(285*n^2-285*n+6)*x^3+(5*n^4-10*n^3-120*n^2+125*n-19)*x^4+(6*n^4-12*n^3-9*n^2+15*n-3)*x^5+(n^4-2*n^3+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[((300*n + (175*n^2 + 95*n)*x - (150*n^3 - 285*n^2 + 33*n)*x^2 - (10*n^4 + 90*n^3 - 125*n^2 + 19*n)*x^3 + (3*n^5 - 12*n^4 - 9*n^3 + 15*n^2 - 3*n)*x^4 + (n^5 - 2*n^4 + n^3)*x^5)*P[n - 1, x] + (125 + 225*x - (175*n^2 - 175*n - 110)*x^2 - (285*n^2 - 285*n + 6)*x^3 + (5*n^4 - 10*n^3 - 120*n^2 + 125*n - 19)*x^4 + (6*n^4 - 12*n^3 - 9*n^2 + 15*n - 3)*x^5 + (n^4 - 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(200*n - 200 + (30*n - 30)*x - (100*n^3 - 300*n^2 + 282*n - 82)*x^2 - (30*n^3 - 90*n^2 + 90*n - 30)*x^3 + (2*n^5 - 10*n^4 + 20*n^3 - 20*n^2 + 10*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] = ((300*n + (175*n^2 + 95*n)*x - (150*n^3 - 285*n^2 + 33*n)*x^2 - (10*n^4 + 90*n^3 - 125*n^2 + 19*n)*x^3 + (3*n^5 - 12*n^4 - 9*n^3 + 15*n^2 - 3*n)*x^4 + (n^5 - 2*n^4 + n^3)*x^5)*v[i - 1] + (125 + 225*x - (175*n^2 - 175*n - 110)*x^2 - (285*n^2 - 285*n + 6)*x^3 + (5*n^4 - 10*n^3 - 120*n^2 + 125*n - 19)*x^4 + (6*n^4 - 12*n^3 - 9*n^2 + 15*n - 3)*x^5 + (n^4 - 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(200*n - 200 + (30*n - 30)*x - (100*n^3 - 300*n^2 + 282*n - 82)*x^2 - (30*n^3 - 90*n^2 + 90*n - 30)*x^3 + (2*n^5 - 10*n^4 + 20*n^3 - 20*n^2 + 10*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