Real-rooted atlas

A130534 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 5\,x + 4\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
26273063
324150309210693
41209422745352216809009
5720673224948458374177815120135135
65040543242419565703397508165236201512002297295
7403204900322533212722422812280257124479366969420166320043648605

All 15 rows

Technical details
Node id
n-7fca598223cf02db · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cd1839a907ad96df

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:       6,       27,        30;
  3:      24,      150,       309,        210;
  4:     120,      942,      2745,       3522,       1680;
  5:     720,     6732,     24948,      45837,      41778,      15120;
  6:    5040,    54324,    241956,     570339,     750816,     523620,     151200;
  7:   40320,   490032,   2533212,    7224228,   12280257,   12447936,    6969420,    1663200;
  8:  362880,  4894128,  28679292,   95416596,  197213049,  259394508,  212100012,   98601840,   19958400;
  9: 3628800, 53658720, 350416584, 1326996756, 3212546238, 5157714717, 5493128724, 3743318556, 1481407920, 259459200;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+5)*x+(n+4)*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[((n + 1 + (2*n + 5)*x + (n + 4)*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] = ((n + 1 + (2*n + 5)*x + (n + 4)*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.