Real-rooted atlas

A143491 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 4\,x - x^{2} - 4\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(5 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 11 + 7\,x + x^{2}\)
\(P_{1}(x) = 50 + 44\,x + 12\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0117119
15044121107
2274296113181702
31764220410452352515274
4130681823610099284543033144712
5109584166984104132345246580721421422568
6102657616817041153032433272977691358011345214407120
71062864018507312137210285700428146561124219325746169863150292720

All 15 rows

Technical details
Node id
n-2ad86c1cd40b71c4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-42f458d827ed1a04

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:         11,          7,          1;
  1:         50,         44,         12,          1;
  2:        274,        296,        113,         18,         1;
  3:       1764,       2204,       1045,        235,        25,        1;
  4:      13068,      18236,      10099,       2845,       430,       33,        1;
  5:     109584,     166984,     104132,      34524,      6580,      721,       42,       1;
  6:    1026576,    1681704,    1153032,     433272,     97769,    13580,     1134,      52,     1;
  7:   10628640,   18507312,   13721028,    5700428,   1465611,   242193,    25746,    1698,    63,    1;
  8:  120543840,  221208912,  175192356,   79081500,  22583315,  4270623,   542283,   45690,  2445,   75,  1;
  9: 1486442880, 2855677824, 2393808120, 1159137496, 361109870, 76141736, 11104632, 1121373, 76890, 3410, 88, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+4*n+4+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 11 + 7*x + x^2, P(1,x) = 50 + 44*x + 12*x^2 + x^3.
%t Mathematica
P[0, x_] := 11 + 7*x + x^2;
P[1, x_] := 50 + 44*x + 12*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 7*x + x^2; v[2] = 50 + 44*x + 12*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 2*x)*v[i - 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

Same fingerprint ?

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.