Real-rooted atlas

A136572 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21024842
34218021648486
42161344288023043847128
51320108003360048000288003840126360
6936095040388800806400864000414720460802624400
7756009172804656960127008001975680016934400677376064512062460720

All 15 rows

Technical details
Node id
n-d468705beb32adb2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       3,         2;
  2:      10,        24,         8;
  3:      42,       180,       216,         48;
  4:     216,      1344,      2880,       2304,        384;
  5:    1320,     10800,     33600,      48000,      28800,        3840;
  6:    9360,     95040,    388800,     806400,     864000,      414720,       46080;
  7:   75600,    917280,   4656960,   12700800,   19756800,    16934400,     6773760,     645120;
  8:  685440,   9676800,  58705920,  198696960,  406425600,   505774080,   361267200,  123863040,   10321920;
  9: 6894720, 111041280, 783820800, 3170119680, 8047226880, 13168189440, 13655900160, 8360755200, 2508226560, 185794560;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (2*n^2+n+(4*n^2+4*n)*x+4*n*x^2)*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^2 + n + (4*n^2 + 4*n)*x + 4*n*x^2)*P[n - 1, x])/(2*n - 1 + 2*x)]];
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^2 + n + (4*n^2 + 4*n)*x + 4*n*x^2)*v[i - 1])/(2*n - 1 + 2*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. [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.