Real-rooted atlas

A109692 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-3\,x - 3\,x^{2} + 2\,n + 10\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
284665119
348444135613653213
43844944237245024439585118881
53840636484201681380228225487814646455587407
6460809365767906176354702968916879611904525065909025318482199
764512015554304160324992915611472312828883263925796887232124600349317832521338307333

All 15 rows

Technical details
Node id
n-f0b30e5b06c6098c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-88408b4d23225179

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:         2,          5;
  2:         8,         46,          65;
  3:        48,        444,        1356,         1365;
  4:       384,       4944,       23724,        50244,         39585;
  5:      3840,      63648,      420168,      1380228,       2254878,        1464645;
  6:     46080,     936576,     7906176,     35470296,      89168796,      119045250,       65909025;
  7:    645120,   15554304,   160324992,    915611472,    3128288832,     6392579688,     7232124600,     3493178325;
  8:  10321920,  288221184,  3514012416,  24429608064,  105904921104,   293106893760,   505661354568,   497050453800,   213083877825;
  9: 185794560, 5900193792, 83139485184, 682199801856, 3591931536288, 12583363643856, 29326280051664, 43837541633592, 38131991113050, 14702787569925;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+(10*n-3)*x+(8*n-3)*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[((2*n + (10*n - 3)*x + (8*n - 3)*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] = ((2*n + (10*n - 3)*x + (8*n - 3)*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. [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.