A219512 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(3\,x^{2} + 6\,n x^{2} - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-x^{2} + x^{3} - 3\,n x^{2} + 3\,n x^{3}\right)\,P_{n-2}'(x) + \left(1 + 6\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	6							7
2	1	24	45						70
3	1	66	423	420					910
4	1	156	2358	7320	4725				14560
5	1	342	10350	70152	133425	62370			276640
6	1	720	39771	501408	1982475	2615760	945945		6086080
7	1	1482	140913	3013164	21194055	56137050	55449135	16216200	152152000

All 15 rows

Technical details

Node id: n-81ff192e090b4a4e · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6d7df8235ce6c6a9

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: 1,    6;
  2: 1,   24,      45;
  3: 1,   66,     423,      420;
  4: 1,  156,    2358,     7320,       4725;
  5: 1,  342,   10350,    70152,     133425,       62370;
  6: 1,  720,   39771,   501408,    1982475,     2615760,      945945;
  7: 1, 1482,  140913,  3013164,   21194055,    56137050,    55449135,    16216200;
  8: 1, 3012,  473868, 16186104,  185065830,   845479620,  1632515220,  1270397520,   310134825;
  9: 1, 6078, 1539072, 80540016, 1410912270, 10180694628, 33128250120, 49343545440, 31375157625, 6547290750;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-6*n-3)*x^2*P(n-2,x) - ((3*n+1)*x^2-(3*n+1)*x^3)*d/dx P(n-2,x) + (1+6*n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 6*n - 3)*x^2*P[n - 2, x] - ((3*n + 1)*x^2 - (3*n + 1)*x^3)*D[P[n - 2, x], x] + (1 + 6*n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], 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] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 6*n - 3)*x^2*v[i - 2] - ((3*n + 1)*x^2 - (3*n + 1)*x^3)*deriv(v[i - 2], x) + (1 + 6*n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.