Real-rooted atlas

A114656 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,x^{2} + 4\,n - 10\,n x + 14\,n x^{2} + 2\,n x^{3} - 10\,n^{2} x + 12\,n^{2} x^{2} - 8\,n^{2} x^{3} + n^{2} x^{4} - 6\,n^{3} x^{2} - 4\,n^{3} x^{3} - n^{3} x^{4} + 6\,n^{4} x^{3} - n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-4 - 20\,x + 4\,x^{2} - 26\,x^{3} - 2\,x^{5} + 20\,n + 16\,n x + 58\,n x^{2} - n x^{3} + 20\,n x^{4} + 3\,n x^{5} - 36\,n^{2} x + 12\,n^{2} x^{2} + 32\,n^{2} x^{3} - 7\,n^{2} x^{4} + 3\,n^{2} x^{5} - 42\,n^{3} x^{2} - 39\,n^{3} x^{3} - 37\,n^{3} x^{4} - 6\,n^{3} x^{5} + 26\,n^{4} x^{3} + 19\,n^{4} x^{4} - n^{4} x^{5} + 5\,n^{5} x^{4} + 3\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(2 + 3\,x^{2} - 4\,x^{4} - x^{6} - 12\,n x + 5\,n x^{2} + 6\,n x^{3} - 6\,n x^{4} + 6\,n x^{5} + n x^{6} - 6\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{2} x^{4} + 2\,n^{2} x^{5} + 2\,n^{2} x^{6} + 12\,n^{3} x^{3} + 2\,n^{3} x^{4} - 12\,n^{3} x^{5} - 2\,n^{3} x^{6} + 2\,n^{4} x^{4} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
31132952
44520413516400
519712161314384253136
690370601071052808753624864
7427940512795155632016125172849198528
820793231196558117520240217875410763087641592448

All 15 rows

Technical details
Node id
n-53a637cf2423ea1c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c431670491fdf571

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:     11,      32,        9;
   4:     45,     204,      135,       16;
   5:    197,    1216,     1314,      384,       25;
   6:    903,    7060,    10710,     5280,      875,      36;
   7:   4279,   40512,    79515,    56320,    16125,    1728,      49;
   8:  20793,  231196,   558117,   520240,   217875,   41076,    3087,     64;
   9: 103049, 1316096,  3775716,  4385024,  2450350,  685440,   91924,   5120,   81;
  10: 518859, 7484580, 24894540, 34708800, 24428250, 9221688, 1856316, 186624, 8019, 100;
%F Formula
The row polynomials satisfy (4*n+4-(10*n^2+10*n)*x-(6*n^3-12*n^2-14*n+4)*x^2+(6*n^4-4*n^3-8*n^2+2*n)*x^3+(n^5-n^4-n^3+n^2)*x^4)*P(n,x) = (20*n-4-(36*n^2-16*n+20)*x-(42*n^3-12*n^2-58*n-4)*x^2+(26*n^4-39*n^3+32*n^2-n-26)*x^3+(5*n^5+19*n^4-37*n^3-7*n^2+20*n)*x^4+(3*n^5-n^4-6*n^3+3*n^2+3*n-2)*x^5)*P(n-1,x) + (2-12*n*x-(6*n^2-5*n-3)*x^2+(12*n^3-2*n^2+6*n)*x^3+(2*n^4+2*n^3+4*n^2-6*n-4)*x^4-(12*n^3-2*n^2-6*n)*x^5-(2*n^4+2*n^3-2*n^2-n+1)*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[((20*n - 4 - (36*n^2 - 16*n + 20)*x - (42*n^3 - 12*n^2 - 58*n - 4)*x^2 + (26*n^4 - 39*n^3 + 32*n^2 - n - 26)*x^3 + (5*n^5 + 19*n^4 - 37*n^3 - 7*n^2 + 20*n)*x^4 + (3*n^5 - n^4 - 6*n^3 + 3*n^2 + 3*n - 2)*x^5)*P[n - 1, x] + (2 - 12*n*x - (6*n^2 - 5*n - 3)*x^2 + (12*n^3 - 2*n^2 + 6*n)*x^3 + (2*n^4 + 2*n^3 + 4*n^2 - 6*n - 4)*x^4 - (12*n^3 - 2*n^2 - 6*n)*x^5 - (2*n^4 + 2*n^3 - 2*n^2 - n + 1)*x^6)*D[P[n - 1, x], x])/(4*n + 4 - (10*n^2 + 10*n)*x - (6*n^3 - 12*n^2 - 14*n + 4)*x^2 + (6*n^4 - 4*n^3 - 8*n^2 + 2*n)*x^3 + (n^5 - n^4 - 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] = ((20*n - 4 - (36*n^2 - 16*n + 20)*x - (42*n^3 - 12*n^2 - 58*n - 4)*x^2 + (26*n^4 - 39*n^3 + 32*n^2 - n - 26)*x^3 + (5*n^5 + 19*n^4 - 37*n^3 - 7*n^2 + 20*n)*x^4 + (3*n^5 - n^4 - 6*n^3 + 3*n^2 + 3*n - 2)*x^5)*v[i - 1] + (2 - 12*n*x - (6*n^2 - 5*n - 3)*x^2 + (12*n^3 - 2*n^2 + 6*n)*x^3 + (2*n^4 + 2*n^3 + 4*n^2 - 6*n - 4)*x^4 - (12*n^3 - 2*n^2 - 6*n)*x^5 - (2*n^4 + 2*n^3 - 2*n^2 - n + 1)*x^6)*deriv(v[i - 1], x))/(4*n + 4 - (10*n^2 + 10*n)*x - (6*n^3 - 12*n^2 - 14*n + 4)*x^2 + (6*n^4 - 4*n^3 - 8*n^2 + 2*n)*x^3 + (n^5 - n^4 - 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