Real-rooted atlas

A085880 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n + n x\right)\,P_{n}(x) = \left(-12 - 42\,x - 30\,x^{2} + 8\,n + 28\,n x + 20\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2257
38405098
4403007506251715
52242240840014000875033614
613441680084000210000262500131250705894
78448126720792000264000049500004950000206250015529668
854912960960720720030030000750750001126125009384375033515625353299947

All 15 rows

Technical details
Node id
n-76cc48d59e2c1eb1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9629d1e9c04dbfbe

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        5;
   3:       8,       40,        50;
   4:      40,      300,       750,        625;
   5:     224,     2240,      8400,      14000,        8750;
   6:    1344,    16800,     84000,     210000,      262500,      131250;
   7:    8448,   126720,    792000,    2640000,     4950000,     4950000,     2062500;
   8:   54912,   960960,   7207200,   30030000,    75075000,   112612500,    93843750,    33515625;
   9:  366080,  7321600,  64064000,  320320000,  1001000000,  2002000000,  2502500000,  1787500000,   558593750;
  10: 2489344, 56010240, 560102400, 3267264000, 12252240000, 30630600000, 51051000000, 54697500000, 34185937500, 9496093750;
%F Formula
The row polynomials satisfy (n+n*x)*P(n,x) = (8*n-12+(28*n-42)*x+(20*n-30)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (28*n - 42)*x + (20*n - 30)*x^2)*P[n - 1, x])/(n + n*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (28*n - 42)*x + (20*n - 30)*x^2)*v[i - 1])/(n + n*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.