Real-rooted atlas

A089231 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + n x + 3\,n^{2} x + n^{2} x^{2} + 2\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(1 - x + 9\,n x - 3\,n x^{2} + n x^{3} + 3\,n^{2} x + 18\,n^{2} x^{2} + 5\,n^{2} x^{3} + 12\,n^{3} x^{2} + 14\,n^{3} x^{3} + n^{4} x^{2} + 10\,n^{4} x^{3} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-x^{2} - x^{3} - x^{4} - 7\,n x^{3} - 5\,n x^{4} - 3\,n^{2} x^{3} - 10\,n^{2} x^{4} - 6\,n^{3} x^{4} - n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1189
21245479
3148324384757
41801080384030008001
51120270019200450002592092941
611685670672003150005443202469601179319
7122410584188160147000050803206914880258048016244649

All 15 rows

Technical details
Node id
n-66a70108663ff2e7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6f246b5e553e9e6e

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: 1,   8;
  2: 1,  24,    54;
  3: 1,  48,   324,    384;
  4: 1,  80,  1080,   3840,     3000;
  5: 1, 120,  2700,  19200,    45000,     25920;
  6: 1, 168,  5670,  67200,   315000,    544320,    246960;
  7: 1, 224, 10584, 188160,  1470000,   5080320,   6914880,    2580480;
  8: 1, 288, 18144, 451584,  5292000,  30481920,  82978560,   92897280,   29393280;
  9: 1, 360, 29160, 967680, 15876000, 137168640, 622339200, 1393459200, 1322697600, 362880000;
%F Formula
The row polynomials satisfy (1+(3*n^2+n-1)*x+(n^4+2*n^3+n^2)*x^2)*P(n,x) = (1+(3*n^2+9*n-1)*x+(n^4+12*n^3+18*n^2-3*n)*x^2+(2*n^5+10*n^4+14*n^3+5*n^2+n)*x^3)*P(n-1,x) - (x^2+(3*n^2+7*n+1)*x^3+(n^4+6*n^3+10*n^2+5*n+1)*x^4)*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[((1 + (3*n^2 + 9*n - 1)*x + (n^4 + 12*n^3 + 18*n^2 - 3*n)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 + 5*n^2 + n)*x^3)*P[n - 1, x] - (x^2 + (3*n^2 + 7*n + 1)*x^3 + (n^4 + 6*n^3 + 10*n^2 + 5*n + 1)*x^4)*D[P[n - 1, x], x])/(1 + (3*n^2 + n - 1)*x + (n^4 + 2*n^3 + n^2)*x^2)]];
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] = ((1 + (3*n^2 + 9*n - 1)*x + (n^4 + 12*n^3 + 18*n^2 - 3*n)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 + 5*n^2 + n)*x^3)*v[i - 1] - (x^2 + (3*n^2 + 7*n + 1)*x^3 + (n^4 + 6*n^3 + 10*n^2 + 5*n + 1)*x^4)*deriv(v[i - 1], x))/(1 + (3*n^2 + n - 1)*x + (n^4 + 2*n^3 + n^2)*x^2)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.