Real-rooted atlas

A185410 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-16\,x + 16\,x^{2} - n - 8\,n x - 112\,n x^{2} + n^{2} + 24\,n^{2} x + 212\,n^{2} x^{2} - 8\,n^{3} x - 152\,n^{3} x^{2} + 36\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(3\,x - 14\,x^{2} + 20\,x^{3} + 4\,n x - 4\,n x^{2} - 128\,n x^{3} - 4\,n^{2} x + 212\,n^{2} x^{3} + 8\,n^{3} x^{2} - 120\,n^{3} x^{3} + 16\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - 6\,x^{3} + 4\,x^{4} + 4\,n x^{2} + 20\,n x^{3} - 24\,n x^{4} - 4\,n^{2} x^{2} - 32\,n^{2} x^{3} + 36\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(1 - 8\,x - n + 24\,n x - 24\,n^{2} x + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 6\,n x - 8\,n x^{2} + 4\,n^{2} x + 16\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2189
31403677
41144540128813
514644644473640010245
6114323157282688332001152150045
7143441900801027840107400020390431362503305
811308810676881057408022514000115223041150912819246850265
9139328575071296795904367846000386106624109060672613580820736971755785
10111805630169368820700160512553720095441552645622060864946716672314150405120022120923825
111354248155628108660305203264082650800193554990720202993348992729455493127723185408156108800123904548214992325
1211062832794081700512014636807418201580003421011138048580031171865637455249408008698678387206017484800075816857629491214691465713925
131318859240230186843866442617608116422588000547226589588481405933201397761470938822737926219205329024097346094080004527909365763615916032692224423300024672525
141956588020287277388286385456307285130436472000812326922028288301945106803200047772575166627843291253242967296952760705792000103739628737536331631788032016992022528160563213048136983606725
Technical details
Node id
n-6cf44b40f7f67fc8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1da90f686ef9e4c9

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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: 1;
  2: 1,     8;
  3: 1,    40,      36;
  4: 1,   144,     540,      128;
  5: 1,   464,    4644,     4736,       400;
  6: 1,  1432,   31572,    82688,     33200,      1152;
  7: 1,  4344,  190080,  1027840,   1074000,    203904,      3136;
  8: 1, 13088, 1067688, 10574080,  22514000,  11522304,   1150912,    8192;
  9: 1, 39328, 5750712, 96795904, 367846000, 386106624, 109060672, 6135808, 20736;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2-n-(8*n^3-24*n^2+8*n+16)*x+(36*n^4-152*n^3+212*n^2-112*n+16)*x^2)*P(n-2,x) - ((4*n^2-4*n-3)*x-(8*n^3-4*n-14)*x^2-(16*n^4-120*n^3+212*n^2-128*n+20)*x^3)*d/dx P(n-2,x) - ((4*n^2-4*n-2)*x^2-(16*n^3-32*n^2+20*n-6)*x^3+(16*n^3-36*n^2+24*n-4)*x^4)*d^2/dx^2 P(n-2,x) - (n-1-(8*n^3-24*n^2+24*n-8)*x)*P(n-1,x) + ((4*n^2-6*n+1)*x-(8*n^3-16*n^2+8*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n - (8*n^3 - 24*n^2 + 8*n + 16)*x + (36*n^4 - 152*n^3 + 212*n^2 - 112*n + 16)*x^2)*P[n - 2, x] - ((4*n^2 - 4*n - 3)*x - (8*n^3 - 4*n - 14)*x^2 - (16*n^4 - 120*n^3 + 212*n^2 - 128*n + 20)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4*n - 2)*x^2 - (16*n^3 - 32*n^2 + 20*n - 6)*x^3 + (16*n^3 - 36*n^2 + 24*n - 4)*x^4)*D[P[n - 2, x], {x, 2}] - (n - 1 - (8*n^3 - 24*n^2 + 24*n - 8)*x)*P[n - 1, x] + ((4*n^2 - 6*n + 1)*x - (8*n^3 - 16*n^2 + 8*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - n - (8*n^3 - 24*n^2 + 8*n + 16)*x + (36*n^4 - 152*n^3 + 212*n^2 - 112*n + 16)*x^2)*v[i - 2] - ((4*n^2 - 4*n - 3)*x - (8*n^3 - 4*n - 14)*x^2 - (16*n^4 - 120*n^3 + 212*n^2 - 128*n + 20)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4*n - 2)*x^2 - (16*n^3 - 32*n^2 + 20*n - 6)*x^3 + (16*n^3 - 36*n^2 + 24*n - 4)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - 1 - (8*n^3 - 24*n^2 + 24*n - 8)*x)*v[i - 1] + ((4*n^2 - 6*n + 1)*x - (8*n^3 - 16*n^2 + 8*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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. [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.