Real-rooted atlas

A196347 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2\,n + 12\,n x + 24\,n x^{2} + 16\,n x^{3} + 12\,n^{2} x + 40\,n^{2} x^{2} + 32\,n^{2} x^{3} + 8\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121618
28128288424
34811525184614412528
43841228882944196608153600445824
53840153600138240049152007680000442368018558720
646080221184024883200117964800276480000318504960144506880884597760
764512036126720487710720289013760090316800001560674304014161674240528482304047499540480

All 15 rows

Technical details
Node id
n-5bfab3ce5853aea4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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:         2,          16;
  2:         8,         128,          288;
  3:        48,        1152,         5184,          6144;
  4:       384,       12288,        82944,        196608,        153600;
  5:      3840,      153600,      1382400,       4915200,       7680000,        4423680;
  6:     46080,     2211840,     24883200,     117964800,     276480000,      318504960,      144506880;
  7:    645120,    36126720,    487710720,    2890137600,    9031680000,    15606743040,    14161674240,     5284823040;
  8:  10321920,   660602880,  10404495360,   73987522560,  289013760000,   665887703040,   906347151360,   676457349120,   214035333120;
  9: 185794560, 13377208320, 240789749760, 1997663109120, 9364045824000, 26968451973120, 48942746173440, 54793045278720, 34673723965440, 9512681472000;
%F Formula
The row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = (2*n+(12*n^2+12*n)*x+(8*n^3+40*n^2+24*n)*x^2+(16*n^3+32*n^2+16*n)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (12*n^2 + 12*n)*x + (8*n^3 + 40*n^2 + 24*n)*x^2 + (16*n^3 + 32*n^2 + 16*n)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*x + 4*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + (12*n^2 + 12*n)*x + (8*n^3 + 40*n^2 + 24*n)*x^2 + (16*n^3 + 32*n^2 + 16*n)*x^3)*v[i - 1])/(1 + (6*n - 2)*x + 4*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.