Real-rooted atlas

A094638 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + n + 6\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
264066112
32425690410562240
412017849896242642217658240
5720138241057604028807639205765761863680
650401190881168864609872017836720277175521787385670819840
7403201134144136380809086886436224768086386233611408227206434588163116072960

All 15 rows

Technical details
Node id
n-8028032e5d6e6b2d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8ef3e5343e1a3f4c

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:       2,         6;
  2:       6,        40,         66;
  3:      24,       256,        904,        1056;
  4:     120,      1784,       9896,       24264,        22176;
  5:     720,     13824,     105760,      402880,       763920,       576576;
  6:    5040,    119088,    1168864,     6098720,     17836720,     27717552,      17873856;
  7:   40320,   1134144,   13638080,    90868864,    362247680,    863862336,    1140822720,     643458816;
  8:  362880,  11860416,  169242624,  1376981056,   6985852544,  22626915904,   45685760256,   52564860864,   26381811456;
  9: 3628800, 135296640, 2238005376, 21554971264, 133199654016, 547618376064, 1497695734144, 2627193580416, 2681801714304, 1213563326976;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(6*n+2)*x+(5*n+1)*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[((n + 1 + (6*n + 2)*x + (5*n + 1)*x^2)*P[n - 1, x])/(1 + 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] = ((n + 1 + (6*n + 2)*x + (5*n + 1)*x^2)*v[i - 1])/(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

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.