Real-rooted atlas

A166973 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 9\,x + 5\,n x\right)\,P_{n-1}(x) + \left(-5\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31719
411843163
51342412591536
61557852910155515307
7181194015470335659331160389
81112404656210281085378546559871775988
911487518161406146149548759064219993335923111062391
10118912846394464565805135567301822344894675584620155391172638727
111235205958569201788643518267662782914781013625076705163298451209323512921519374
1212863140517013154873555573715611255787401121877461168022316452565569172153072559411153221709973
1313424599131459451184894702491636092284305795841638218545924165832978453627549625706268149624143535646711037320865141
141403651435491486263202940734970436211812947889610451698216964675951517237911100421731558667608973616899318151182612138803121517178350762
15146989726913949454326872619457039602419467357738529783106195237076215989813130685743665477197197849460976945581061728947591862105101156728328191472862758184789
Technical details
Node id
n-824a6562970843d2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8778b62c1dd7b5cc

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   7,     1;
   4: 1,  18,    43,      1;
   5: 1,  34,   241,    259,       1;
   6: 1,  55,   785,   2910,    1555,        1;
   7: 1,  81,  1940,  15470,   33565,     9331,        1;
   8: 1, 112,  4046,  56210,  281085,   378546,    55987,        1;
   9: 1, 148,  7518, 161406, 1461495,  4875906,  4219993,   335923,       1;
  10: 1, 189, 12846, 394464, 5658051, 35567301, 82234489, 46755846, 2015539, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(5*n-9)*x)*P(n-1,x) - 5*x^2*d/dx 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[(1 + (5*n - 9)*x)*P[n - 1, x] - 5*x^2*D[P[n - 1, x], 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] = (1 + (5*n - 9)*x)*v[i - 1] - 5*x^2*deriv(v[i - 1], 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.