Real-rooted atlas

A038331 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(289 + 425\,x + 100\,x^{2} + 255\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4913 + 13005\,x + 7225\,x^{2} + 1125\,x^{3} + 4335\,n x + 3825\,n x^{2} + 750\,n x^{3} + 425\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 680\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02896802251194
149131734011475200035728
283521393040390150136000156251018336
314198578352100110542505780000132812511250028046832
4241375691703828402818833751965200006773437511475000765625752898784
5410338673337925966067088243255846470000268679687568276250091109375500000019810561408
66975757441656541876801520666847001590239840009135109375030951900000619543750068000000031640625512930685696
71185878764971255636339380332374325130040551115920002795343468750118391017500031596731250052020000000484101562519531250013105356343552

All 15 rows

Technical details
Node id
n-6ffa5f86ce5bd2a4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-061a3a6b806e5c47

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 2)
  • 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:            289,             680,              225;
  1:           4913,           17340,            11475,             2000;
  2:          83521,          393040,           390150,           136000,            15625;
  3:        1419857,         8352100,         11054250,          5780000,          1328125,           112500;
  4:       24137569,       170382840,        281883375,        196520000,         67734375,         11475000,          765625;
  5:      410338673,      3379259660,       6708824325,       5846470000,       2686796875,        682762500,        91109375,         5000000;
  6:     6975757441,     65654187680,     152066684700,     159023984000,      91351093750,      30951900000,      6195437500,       680000000,       31640625;
  7:   118587876497,   1255636339380,    3323743251300,    4055111592000,    2795343468750,    1183910175000,    315967312500,     52020000000,     4841015625,     195312500;
  8:  2015993900449,  23717575299400,   70629544090125,   98481281520000,   79201398281250,   40252945950000,  13428610781250,   2947800000000,   411486328125,   33203125000,   1181640625;
  9: 34271896307633, 443518658098780, 1467524971650375, 2301999955530000, 2115808782656250, 1254550148775000, 502230043218750, 137809650000000, 25649314453125, 3104492187500, 220966796875, 7031250000;
%F Formula
The row polynomials satisfy (289+(255*n+425)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (4913+(4335*n+13005)*x+(425*n^2+3825*n+7225)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 680*x + 225*x^2.
%t Mathematica
P[0, x_] := 289 + 680*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4913 + (4335*n + 13005)*x + (425*n^2 + 3825*n + 7225)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(289 + (255*n + 425)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 680*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (4335*n + 13005)*x + (425*n^2 + 3825*n + 7225)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(289 + (255*n + 425)*x + (25*n^2 + 100*n + 100)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.