Real-rooted atlas

A033282 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(32 - 24\,x + 16\,x^{2} + 16\,n - 12\,n x - 32\,n x^{2} - 6\,n x^{3} - 60\,n^{2} x^{2} - 9\,n^{2} x^{3} + 2\,n^{2} x^{4} - 20\,n^{3} x^{2} - 3\,n^{3} x^{3} + 5\,n^{3} x^{4} + 4\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(96 + 72\,x - 96\,x^{2} - 112\,x^{3} - 24\,x^{4} + 4\,x^{5} + 48\,n + 148\,n x - 216\,n x^{2} - 322\,n x^{3} - 60\,n x^{4} + 18\,n x^{5} + 56\,n^{2} x - 204\,n^{2} x^{2} - 287\,n^{2} x^{3} - 42\,n^{2} x^{4} + 32\,n^{2} x^{5} - 60\,n^{3} x^{2} - 93\,n^{3} x^{3} + 3\,n^{3} x^{4} + 28\,n^{3} x^{5} - 8\,n^{4} x^{3} + 12\,n^{4} x^{4} + 12\,n^{4} x^{5} + 3\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(16 - 76\,x^{2} - 96\,x^{3} - 38\,x^{4} + 2\,x^{6} - 112\,n x^{2} - 192\,n x^{3} - 76\,n x^{4} + 12\,n x^{5} + 8\,n x^{6} - 56\,n^{2} x^{2} - 96\,n^{2} x^{3} - 22\,n^{2} x^{4} + 30\,n^{2} x^{5} + 12\,n^{2} x^{6} + 16\,n^{3} x^{4} + 24\,n^{3} x^{5} + 8\,n^{3} x^{6} + 4\,n^{4} x^{4} + 6\,n^{4} x^{5} + 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
2116045116
3453725672241208
4197218450404032105012503
5903125603888047040247504752128885
6427971604278235451440358875138996210211324450
72079340670019022853874640412912524144127357359152013575210

All 15 rows

Technical details
Node id
n-db828bb4d1dd2f0c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6b2b9a357366bd53

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 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:      3,        8;
  2:     11,       60,       45;
  3:     45,      372,      567,       224;
  4:    197,     2184,     5040,      4032,      1050;
  5:    903,    12560,    38880,     47040,     24750,      4752;
  6:   4279,    71604,   278235,    451440,    358875,    138996,     21021;
  7:  20793,   406700,  1902285,   3874640,   4129125,   2414412,    735735,    91520;
  8: 103049,  2306832, 12618144,  30942912,  41391350,  32432400,  14910896,  3734016,   393822;
  9: 518859, 13081752, 81931824, 235015872, 378500850, 372206016, 229228272, 86561280, 18366426, 1679600;
%F Formula
The row polynomials satisfy (16*n+32-(12*n+24)*x-(20*n^3+60*n^2+32*n-16)*x^2-(3*n^3+9*n^2+6*n)*x^3+(n^5+4*n^4+5*n^3+2*n^2)*x^4)*P(n,x) = (48*n+96+(56*n^2+148*n+72)*x-(60*n^3+204*n^2+216*n+96)*x^2-(8*n^4+93*n^3+287*n^2+322*n+112)*x^3+(3*n^5+12*n^4+3*n^3-42*n^2-60*n-24)*x^4+(2*n^5+12*n^4+28*n^3+32*n^2+18*n+4)*x^5)*P(n-1,x) + (16-(56*n^2+112*n+76)*x^2-(96*n^2+192*n+96)*x^3+(4*n^4+16*n^3-22*n^2-76*n-38)*x^4+(6*n^4+24*n^3+30*n^2+12*n)*x^5+(2*n^4+8*n^3+12*n^2+8*n+2)*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[((48*n + 96 + (56*n^2 + 148*n + 72)*x - (60*n^3 + 204*n^2 + 216*n + 96)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (3*n^5 + 12*n^4 + 3*n^3 - 42*n^2 - 60*n - 24)*x^4 + (2*n^5 + 12*n^4 + 28*n^3 + 32*n^2 + 18*n + 4)*x^5)*P[n - 1, x] + (16 - (56*n^2 + 112*n + 76)*x^2 - (96*n^2 + 192*n + 96)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (6*n^4 + 24*n^3 + 30*n^2 + 12*n)*x^5 + (2*n^4 + 8*n^3 + 12*n^2 + 8*n + 2)*x^6)*D[P[n - 1, x], x])/(16*n + 32 - (12*n + 24)*x - (20*n^3 + 60*n^2 + 32*n - 16)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*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] = ((48*n + 96 + (56*n^2 + 148*n + 72)*x - (60*n^3 + 204*n^2 + 216*n + 96)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (3*n^5 + 12*n^4 + 3*n^3 - 42*n^2 - 60*n - 24)*x^4 + (2*n^5 + 12*n^4 + 28*n^3 + 32*n^2 + 18*n + 4)*x^5)*v[i - 1] + (16 - (56*n^2 + 112*n + 76)*x^2 - (96*n^2 + 192*n + 96)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (6*n^4 + 24*n^3 + 30*n^2 + 12*n)*x^5 + (2*n^4 + 8*n^3 + 12*n^2 + 8*n + 2)*x^6)*deriv(v[i - 1], x))/(16*n + 32 - (12*n + 24)*x - (20*n^3 + 60*n^2 + 32*n - 16)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*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 - 1, v[i])))};
T(9)

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