Real-rooted atlas

A059364 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 8\,n x - 16\,n x^{2} + 4\,n^{2} + 24\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} - 16\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 8\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
25910832199
3593199219043844873
470173522461040400646144149489
596087649244166744019712649917441228805498659
6149861112802868440212007673030469259264283484162949120235609783
72626294527218145611789497922741028864363507942426627891209205678088257536011519434769

All 15 rows

Technical details
Node id
n-2eba56c0d02ad420 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1272df06034c859c

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:           7,            4;
  2:          59,          108,           32;
  3:         593,         1992,         1904,           384;
  4:        7017,        35224,        61040,         40064,          6144;
  5:       96087,       649244,      1667440,       1971264,        991744,         122880;
  6:     1498611,     12802868,     44021200,      76730304,      69259264,       28348416,       2949120;
  7:    26262945,    272181456,   1178949792,    2741028864,    3635079424,     2662789120,     920567808,      82575360;
  8:   511334865,   6243048432,  32719081632,   95576972288,  168624232704,   181031473152,  111820193792,   33504362496,    2642411520;
  9: 10955901735, 154193491188, 950203408416, 3351881353856, 7415316188928, 10570027889664, 9551245967360, 5107252396032, 1351489683456, 95126814720;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(16*n^3-24*n^2+8*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(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] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(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. [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.