Real-rooted atlas

A166340 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-8\,n x + 16\,n x^{2} - 4\,n^{2} x + 25\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 17\,x^{2} - 23\,x^{3} - 2\,n x + 16\,n x^{2} - 29\,n x^{3} + 4\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 9\,x^{3} - 7\,x^{4} - n x^{2} + 12\,n x^{3} - 11\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 4\,x + n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 8\,x^{2} + 2\,n x - 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 32\,x + 9\,x^{2}\)
\(P_{2}(x) = 1 + 76\,x + 171\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1132942
217617116264
311681026672251892
413564590816022253615368
51736177756528049375662449139840
6115006326141912065487525304418375641410240
7130322134082351264654875052903441161888485128115617280
816100694800120608005478685078893064369362004940800123525100188442240

All 15 rows

Technical details
Node id
n-60b98be0191f169e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8f2ceadb5f7321ff

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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,    32,       9;
   2: 1,    76,     171,        16;
   3: 1,   168,    1026,       672,         25;
   4: 1,   356,    4590,      8160,       2225,         36;
   5: 1,   736,   17775,     65280,      49375,       6624,          49;
   6: 1,  1500,   63261,    419120,     654875,     253044,       18375,         64;
   7: 1,  3032,  213408,   2351264,    6548750,    5290344,     1161888,      48512,         81;
   8: 1,  6100,  694800,  12060800,   54786850,   78893064,    36936200,    4940800,     123525,      100;
   9: 1, 12240, 2207925,  58124800,  405849250,  946716768,   795464530,  232499200,   19871325,   306000,    121;
  10: 1, 24524, 6899175, 267826000, 2755806250, 9771261048, 13299771982, 7054864000, 1355869125, 76657500, 741851, 144;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -((4*n^2+8*n)*x-(9*n^3+25*n^2+16*n)*x^2)*P(n-2,x) - ((2*n+4)*x-(4*n^2+16*n+17)*x^2-(4*n^3-2*n^2-29*n-23)*x^3)*d/dx P(n-2,x) - ((n+2)*x^2-(4*n^2+12*n+9)*x^3+(4*n^2+11*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (n+1+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+3)*x-(4*n^2+12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 32*x + 9*x^2, P(2,x) = 1 + 76*x + 171*x^2 + 16*x^3.
%t Mathematica
P[1, x_] := 1 + 32*x + 9*x^2;
P[2, x_] := 1 + 76*x + 171*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*P[n - 2, x] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 1 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 32*x + 9*x^2; v[2] = 1 + 76*x + 171*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*v[i - 2] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 1 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.