Real-rooted atlas

A028326 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 6\,x + 18\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 36\,x + 54\,x^{2} + 27\,x^{3} + 36\,n x + 90\,n x^{2} + 54\,n x^{3} + 18\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
142428
2896162266
3162889728642140
43276838886912405015650
564192012960345604050017496107500
612846083888013824024300020995271442706250
7256107521088644838401134000146966410001882799364487500

All 15 rows

Technical details
Node id
n-a284f6ab56362b01 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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:    2;
  1:    4,    24;
  2:    8,    96,    162;
  3:   16,   288,    972,     864;
  4:   32,   768,   3888,    6912,     4050;
  5:   64,  1920,  12960,   34560,    40500,    17496;
  6:  128,  4608,  38880,  138240,   243000,   209952,    71442;
  7:  256, 10752, 108864,  483840,  1134000,  1469664,  1000188,   279936;
  8:  512, 24576, 290304, 1548288,  4536000,  7838208,  8001504,  4478976,  1062882;
  9: 1024, 55296, 746496, 4644864, 16329600, 35271936, 48009024, 40310784, 19131876, 3936600;
%F Formula
The row polynomials satisfy (4+(18*n-6)*x+9*n^2*x^2)*P(n,x) = (8+(36*n+36)*x+(18*n^2+90*n+54)*x^2+(27*n^2+54*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + (36*n + 36)*x + (18*n^2 + 90*n + 54)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(4 + (18*n - 6)*x + 9*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] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (36*n + 36)*x + (18*n^2 + 90*n + 54)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(4 + (18*n - 6)*x + 9*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

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.