Real-rooted atlas

A204420 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
22736972
3405121510352252880
4850545360812705544011025201600
5229635191362558684508079750478831589302521772800
675779559093546042520747597671420011356641456088559401080560253353011200
729554024548271573353195118426510995191707520935560145521567262216510711083883518261468225697426329600

All 15 rows

Technical details
Node id
n-4643327b1b00e955 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fe2eac7fb802bf9f

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:            3,              1;
  2:           27,             36,               9;
  3:          405,           1215,            1035,              225;
  4:         8505,          45360,           81270,            55440,            11025;
  5:       229635,        1913625,         5868450,          8079750,          4788315,            893025;
  6:      7577955,       90935460,       425207475,        976714200,       1135664145,         608855940,         108056025;
  7:    295540245,     4827157335,     31951184265,     109951917075,     209355601455,      215672622165,      107110838835,       18261468225;
  8:  13299311025,   283718635200,   2523913692300,   12136852728000,   34160183407350,    56810278324800,    53346327734700,    24921704808000,    4108830350625;
  9: 678264862275, 18313151281425, 210714283880100, 1348390546202700, 5249719792166850, 12769617199288950, 19138833150336900, 16688095660536300, 7411923037393875, 1187451971330625;
%F Formula
The row polynomials satisfy P(n,x) = (6*n-3+(4*n^2-4*n+1)*x)*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[(6*n - 3 + (4*n^2 - 4*n + 1)*x)*P[n - 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] = (6*n - 3 + (4*n^2 - 4*n + 1)*x)*v[i - 1]); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.