Real-rooted atlas

A111125 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-300 - 120\,x + 312\,x^{2} + 120\,x^{3} - 12\,x^{4} + 200\,n + 80\,n x - 658\,n x^{2} - 242\,n x^{3} + 44\,n x^{4} + 450\,n^{2} x^{2} + 162\,n^{2} x^{3} - 63\,n^{2} x^{4} - 100\,n^{3} x^{2} - 36\,n^{3} x^{3} + 44\,n^{3} x^{4} - 15\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-200 - 130\,x + 48\,x^{2} + 46\,x^{3} + 8\,x^{4} + 300\,n - 5\,n x - 452\,n x^{2} - 279\,n x^{3} - 44\,n x^{4} + 175\,n^{2} x + 550\,n^{2} x^{2} + 318\,n^{2} x^{3} + 34\,n^{2} x^{4} - 4\,n^{2} x^{5} - 150\,n^{3} x^{2} - 79\,n^{3} x^{3} + 19\,n^{3} x^{4} + 8\,n^{3} x^{5} - 10\,n^{4} x^{3} - 20\,n^{4} x^{4} - 5\,n^{4} x^{5} + 3\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 300\,x + 160\,x^{2} - 64\,x^{3} - 57\,x^{4} - 8\,x^{5} + 350\,n x^{2} + 600\,n x^{3} + 286\,n x^{4} + 36\,n x^{5} - 175\,n^{2} x^{2} - 300\,n^{2} x^{3} - 123\,n^{2} x^{4} + 6\,n^{2} x^{5} + 4\,n^{2} x^{6} - 20\,n^{3} x^{4} - 24\,n^{3} x^{5} - 4\,n^{3} x^{6} + 5\,n^{4} x^{4} + 6\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2448
31128948
4291249016259
576460540208251309
6199156025651568400366328
7521501610665905636256844929616
81364155724071644560246257236107864135215

All 15 rows

Technical details
Node id
n-3221e64094f10734 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e9dc029a14cf0a23

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:    4,      4;
   3:   11,     28,      9;
   4:   29,    124,     90,     16;
   5:   76,    460,    540,    208,     25;
   6:  199,   1560,   2565,   1568,    400,     36;
   7:  521,   5016,  10665,   9056,   3625,    684,     49;
   8: 1364,  15572,  40716,  44560,  24625,   7236,   1078,    64;
   9: 3571,  47156, 146520, 197008, 139875,  56484,  13034,  1600,   81;
  10: 9349, 140180, 504945, 806944, 702825, 363636, 114905, 21760, 2268, 100;
%F Formula
The row polynomials satisfy (200*n-300+(80*n-120)*x-(100*n^3-450*n^2+658*n-312)*x^2-(36*n^3-162*n^2+242*n-120)*x^3+(2*n^5-15*n^4+44*n^3-63*n^2+44*n-12)*x^4)*P(n,x) = (300*n-200+(175*n^2-5*n-130)*x-(150*n^3-550*n^2+452*n-48)*x^2-(10*n^4+79*n^3-318*n^2+279*n-46)*x^3+(3*n^5-20*n^4+19*n^3+34*n^2-44*n+8)*x^4+(n^5-5*n^4+8*n^3-4*n^2)*x^5)*P(n-1,x) + (125+300*x-(175*n^2-350*n-160)*x^2-(300*n^2-600*n+64)*x^3+(5*n^4-20*n^3-123*n^2+286*n-57)*x^4+(6*n^4-24*n^3+6*n^2+36*n-8)*x^5+(n^4-4*n^3+4*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 - 200 + (175*n^2 - 5*n - 130)*x - (150*n^3 - 550*n^2 + 452*n - 48)*x^2 - (10*n^4 + 79*n^3 - 318*n^2 + 279*n - 46)*x^3 + (3*n^5 - 20*n^4 + 19*n^3 + 34*n^2 - 44*n + 8)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*P[n - 1, x] + (125 + 300*x - (175*n^2 - 350*n - 160)*x^2 - (300*n^2 - 600*n + 64)*x^3 + (5*n^4 - 20*n^3 - 123*n^2 + 286*n - 57)*x^4 + (6*n^4 - 24*n^3 + 6*n^2 + 36*n - 8)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*D[P[n - 1, x], x])/(200*n - 300 + (80*n - 120)*x - (100*n^3 - 450*n^2 + 658*n - 312)*x^2 - (36*n^3 - 162*n^2 + 242*n - 120)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*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 - 200 + (175*n^2 - 5*n - 130)*x - (150*n^3 - 550*n^2 + 452*n - 48)*x^2 - (10*n^4 + 79*n^3 - 318*n^2 + 279*n - 46)*x^3 + (3*n^5 - 20*n^4 + 19*n^3 + 34*n^2 - 44*n + 8)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*v[i - 1] + (125 + 300*x - (175*n^2 - 350*n - 160)*x^2 - (300*n^2 - 600*n + 64)*x^3 + (5*n^4 - 20*n^3 - 123*n^2 + 286*n - 57)*x^4 + (6*n^4 - 24*n^3 + 6*n^2 + 36*n - 8)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*deriv(v[i - 1], x))/(200*n - 300 + (80*n - 120)*x - (100*n^3 - 450*n^2 + 658*n - 312)*x^2 - (36*n^3 - 162*n^2 + 242*n - 120)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*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

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