Real-rooted atlas

A145324 → basis A013609 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2344882
32667845761626
421961002415216768035116
519692122872286944297216115200841924
61928161534352487586477340166122880193536022395288
72057904199792808070722417362457620978624013497753636126720657259480

All 15 rows

Technical details
Node id
n-b045d4d1d6148fbd · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e7900e406f467c81

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:         0;
  1:         4;
  2:        34,         48;
  3:       266,        784,         576;
  4:      2196,      10024,       15216,        7680;
  5:     19692,     122872,      286944,      297216,       115200;
  6:    192816,    1534352,     4875864,     7734016,      6122880,      1935360;
  7:   2057904,   19979280,    80707224,   173624576,    209786240,    134977536,     36126720;
  8:  23844960,  273877344,  1346512800,  3673219136,   6004415360,   5881158144,   3195795456,   743178240;
  9: 298582560, 3966449184, 23027961120, 76312418496, 157879047360, 208798708224, 172380773376, 81221713920, 16721510400;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(4*n^3+8*n^2+4*n)*x+(4*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(4*n^2+4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x + (4*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x + (4*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*v[i - 1])/(n)); 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.