Real-rooted atlas

A130534 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-1 - 2\,x - x^{2} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3} - n^{2} - 4\,n^{2} x - 6\,n^{2} x^{2} - 4\,n^{2} x^{3} - n^{2} x^{4}\right)\,P_{n-2}(x) + \left(-1 - x + x^{2} + x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
25151131
3177410650247
4744168698002742433
53942694731698696616176428653
6248419916661491165511149396017213068393279
7181081660886496991405564181681614034926001041095846169455

All 15 rows

Technical details
Node id
n-246aa30d8697760e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cf35f6a057a254f4

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,        3;
  2:       5,       15,       11;
  3:      17,       74,      106,        50;
  4:      74,      416,      869,       800,       274;
  5:     394,     2694,     7316,      9869,      6616,      1764;
  6:    2484,    19916,    66149,    116551,    114939,     60172,     13068;
  7:   18108,   166088,   649699,   1405564,   1816816,   1403492,    600104,    109584;
  8:  149904,  1545064,  6938112,  17734032,  28227569,  28657392,  18125232,   6530904,  1026576;
  9: 1389456, 15877080, 80339964, 236336792, 445512222, 558203799, 464947792, 248293572, 77150448, 10628640;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n^2-2*n+1+(4*n^2-6*n+2)*x+(6*n^2-6*n+1)*x^2+(4*n^2-2*n)*x^3+n^2*x^4)*P(n-2,x) + (2*n-1+(6*n-1)*x+(6*n+1)*x^2+(2*n+1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 + (4*n^2 - 6*n + 2)*x + (6*n^2 - 6*n + 1)*x^2 + (4*n^2 - 2*n)*x^3 + n^2*x^4)*P[n - 2, x] + (2*n - 1 + (6*n - 1)*x + (6*n + 1)*x^2 + (2*n + 1)*x^3)*P[n - 1, x])/(1 + 2*x + 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; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 + (4*n^2 - 6*n + 2)*x + (6*n^2 - 6*n + 1)*x^2 + (4*n^2 - 2*n)*x^3 + n^2*x^4)*v[i - 2] + (2*n - 1 + (6*n - 1)*x + (6*n + 1)*x^2 + (2*n + 1)*x^3)*v[i - 1])/(1 + 2*x + 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

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.