Real-rooted atlas

A130534 → D²∘x (shift +1) → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{3} + 3\,n x^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n-3}(x) + \left(-1 - 3\,x - 3\,x^{2} - 3\,n - 6\,n x - 3\,n^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)
\(P_{2}(x) = 35 + 30\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
23530671
32252559010580
4162422051050210155104
513132203071176032204202148860
611812420185213469445360819075628509004
7117270021710401615950632730141750182701260365753736

All 15 rows

Technical details
Node id
n-96653194bcf0694e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1db1b1b28f935ec6

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         6,         3;
  2:        35,        30,         6;
  3:       225,       255,        90,        10;
  4:      1624,      2205,      1050,       210,       15;
  5:     13132,     20307,     11760,      3220,      420,      21;
  6:    118124,    201852,    134694,     45360,     8190,     756,     28;
  7:   1172700,   2171040,   1615950,    632730,   141750,   18270,   1260,    36;
  8:  12753576,  25228500,  20501580,   9020550,  2366595,  381150,  36960,  1980,   45;
  9: 150917976, 315774228, 275974380, 133395350, 39563370, 7505883, 914760, 69300, 2970, 55;
%F Formula
The row polynomials satisfy P(n,x) = (n^3+3*n^2*x+3*n*x^2+x^3)*P(n-3,x) - (3*n^2+3*n+1+(6*n+3)*x+3*x^2)*P(n-2,x) + (3*n+3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 3*x, P(2,x) = 35 + 30*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[2, x_] := 35 + 30*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(n^3 + 3*n^2*x + 3*n*x^2 + x^3)*P[n - 3, x] - (3*n^2 + 3*n + 1 + (6*n + 3)*x + 3*x^2)*P[n - 2, x] + (3*n + 3 + 3*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; v[2] = 6 + 3*x; v[3] = 35 + 30*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (n^3 + 3*n^2*x + 3*n*x^2 + x^3)*v[i - 3] - (3*n^2 + 3*n + 1 + (6*n + 3)*x + 3*x^2)*v[i - 2] + (3*n + 3 + 3*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. [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.