Real-rooted atlas

A196347 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2\,n + 8\,n x + 6\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
284872128
348432129612963072
4384460820736414723110498304
5384057600345600103680015552009331203932160
646080829440622080024883200559872006718464033592320188743680
764512013547520121927680609638400182891520032920473603292047360141087744010569646080

All 15 rows

Technical details
Node id
n-816ba7e3126bda81 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f9762973ec1089d7

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,          6;
  2:         8,         48,          72;
  3:        48,        432,        1296,         1296;
  4:       384,       4608,       20736,        41472,         31104;
  5:      3840,      57600,      345600,      1036800,       1555200,        933120;
  6:     46080,     829440,     6220800,     24883200,      55987200,      67184640,       33592320;
  7:    645120,   13547520,   121927680,    609638400,    1828915200,    3292047360,     3292047360,     1410877440;
  8:  10321920,  247726080,  2601123840,  15606743040,   58525286400,  140460687360,   210691031040,   180592312320,    67722117120;
  9: 185794560, 5016453120, 60197437440, 421382062080, 1896219279360, 5688657838080, 11377315676160, 14627977297920, 10970982973440, 3656994324480;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+8*n*x+6*n*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 + 8*n*x + 6*n*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 + 8*n*x + 6*n*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

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.