Real-rooted atlas

A225467 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 - 5\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
31640965
46433631627743
5256230449602320819921
6102414080544006384016564243150151
740967987248512010713607684961169207292526593
816384430080377753613502720191443848921136821356218746615783

All 15 rows

Technical details
Node id
n-9347079e8aa5e773 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2498353059445bbe

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:      4,        3;
   3:     16,       40,         9;
   4:     64,      336,       316,         27;
   5:    256,     2304,      4960,       2320,         81;
   6:   1024,    14080,     54400,      63840,      16564,        243;
   7:   4096,    79872,    485120,    1071360,     768496,     116920,        729;
   8:  16384,   430080,   3777536,   13502720,   19144384,    8921136,     821356,       2187;
   9:  65536,  2228224,  26722304,  140894208,  333129216,  322850304,  101417920,    5758240,     6561;
  10: 262144, 11206656, 175964160, 1285079040, 4573083648, 7620856320, 5248426240, 1138630080, 40333924, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (4+(4*n-5)*x)*P(n-1,x) - 4*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[(4 + (4*n - 5)*x)*P[n - 1, x] - 4*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] = (4 + (4*n - 5)*x)*v[i - 1] - 4*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.