A049403 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 4\,x + 5\,n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(12\,x - 6\,x^{2} - 6\,n + 13\,n x - 2\,n x^{2} - 5\,n^{2} + 3\,n^{2} x - n^{3}\right)\,P_{n-1}(x) + \left(12\,x - 12\,x^{2} + 10\,n x - 4\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6\,x + 3\,x^{2}\)

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	6	3								9
1	6	18	4							28
2	0	45	40	5						90
3	0	45	180	75	6					306
4	0	0	420	525	126	7				1078
5	0	0	420	2100	1260	196	8			3984
6	0	0	0	4725	7560	2646	288	9		15228
7	0	0	0	4725	28350	22050	5040	405	10	60580

All 15 rows

Technical details

Node id: n-60623139986cb3c3 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-dbce68f6ba2c0d6a

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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: 0, 6,  3;
  1: 0, 6, 18,   4;
  2: 0, 0, 45,  40,    5;
  3: 0, 0, 45, 180,   75,     6;
  4: 0, 0,  0, 420,  525,   126,      7;
  5: 0, 0,  0, 420, 2100,  1260,    196,      8;
  6: 0, 0,  0,   0, 4725,  7560,   2646,    288,      9;
  7: 0, 0,  0,   0, 4725, 28350,  22050,   5040,    405,    10;
  8: 0, 0,  0,   0,    0, 62370, 121275,  55440,   8910,   550,  11;
  9: 0, 0,  0,   0,    0, 62370, 436590, 415800, 124740, 14850, 726, 12;

%F Formula

The row polynomials satisfy (n^2+5*n+6-(2*n+4)*x)*P(n,x) = -(n^3+5*n^2+6*n-(3*n^2+13*n+12)*x+(2*n+6)*x^2)*P(n-1,x) + ((2*n^2+10*n+12)*x-(4*n+12)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6*x + 3*x^2.

%t Mathematica

P[0, x_] := 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 5*n^2 + 6*n - (3*n^2 + 13*n + 12)*x + (2*n + 6)*x^2)*P[n - 1, x] + ((2*n^2 + 10*n + 12)*x - (4*n + 12)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n + 6 - (2*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 5*n^2 + 6*n - (3*n^2 + 13*n + 12)*x + (2*n + 6)*x^2)*v[i - 1] + ((2*n^2 + 10*n + 12)*x - (4*n + 12)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n + 6 - (2*n + 4)*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A122848 → D

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.