Real-rooted atlas

A034870 → D → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1426
2163624480
336150240180606672
46439210081400112050411284608
51008102880588075606300336010801801028160
61441452660017820316803880833264198007920198026412159744
7196236613104440441001001621621921921681681081085005016016327636414860160

All 15 rows

Technical details
Node id
n-4b7013a809f06d5e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6fde533e0acf53df

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   4,    2;
  2:  16,   36,    24,      4;
  3:  36,  150,   240,    180,     60,       6;
  4:  64,  392,  1008,   1400,   1120,     504,     112,       8;
  5: 100,  810,  2880,   5880,   7560,    6300,    3360,    1080,     180,      10;
  6: 144, 1452,  6600,  17820,  31680,   38808,   33264,   19800,    7920,    1980,     264,      12;
  7: 196, 2366, 13104,  44044, 100100,  162162,  192192,  168168,  108108,   50050,   16016,    3276,    364,     14;
  8: 256, 3600, 23520,  94640, 262080,  528528,  800800,  926640,  823680,  560560,  288288,  109200,  29120,   5040,   480,   16;
  9: 324, 5202, 39168, 183600, 599760, 1447992, 2673216, 3850704, 4375800, 3938220, 2800512, 1559376, 668304, 214200, 48960, 7344, 612, 18;
%F Formula
The row polynomials satisfy (2*n^2-4*n+2+(n-1)*x)*P(n,x) = (2*n^2+(4*n^2+n)*x+(2*n^2+2*n)*x^2+n*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + (4*n^2 + n)*x + (2*n^2 + 2*n)*x^2 + n*x^3)*P[n - 1, x])/(2*n^2 - 4*n + 2 + (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] = 0; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + (4*n^2 + n)*x + (2*n^2 + 2*n)*x^2 + n*x^3)*v[i - 1])/(2*n^2 - 4*n + 2 + (n - 1)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.