Real-rooted atlas

A204420 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 - x - 2\,x^{2} - 4\,n - 2\,n x + 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
295472135
32251710396028808775
411025995403137404183202016001044225
5893025925344036163260677678406150816021772800197358525
610805602512571920905800784220137690506801787873472012106765440335301120054273594375
7182614682252349411164101241828487900353353268268058854687091205764820181120308486610432069742632960020461145079375

All 15 rows

Technical details
Node id
n-3df856e22293eef3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e426df4ea063f288

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:                1,                 4;
  2:                9,                54,                 72;
  3:              225,              1710,               3960,               2880;
  4:            11025,             99540,             313740,             418320,              201600;
  5:           893025,           9253440,           36163260,           67767840,            61508160,            21772800;
  6:        108056025,        1257192090,         5800784220,        13769050680,         17878734720,         12106765440,          3353011200;
  7:      18261468225,      234941116410,      1241828487900,      3533532682680,       5885468709120,       5764820181120,       3084866104320,        697426329600;
  8:    4108830350625,    57792347613000,    342845511208200,   1130338545336000,    2278284283875600,    2886161092214400,    2250596322374400,     989834772326400,    188305108992000;
  9: 1187451971330625, 18098990779369500, 118731750927589800, 443235313412892000, 1042739263454288400, 1608717212167665600, 1631717108519097600, 1051264998809625600, 390963999089664000, 64023737057280000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n^2-4*n+1+(8*n^2-2*n-1)*x+(4*n^2+2*n-2)*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[((4*n^2 - 4*n + 1 + (8*n^2 - 2*n - 1)*x + (4*n^2 + 2*n - 2)*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] = ((4*n^2 - 4*n + 1 + (8*n^2 - 2*n - 1)*x + (4*n^2 + 2*n - 2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.