A094262 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6\,x^{2} - 12\,x^{3} - 6\,x^{4}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 6\,x^{3} - 6\,x^{4} - 2\,x^{5}\right)\,P_{n-2}'(x) + \left(1 + 8\,x + 7\,x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	8	9													18
2	1	24	108	160	75											368
3	1	56	549	1984	3275	2520	735									9120
4	1	120	2160	14240	45750	80136	78400	40320	8505							269632
5	1	248	7569	80960	424750	1267776	2306038	2610944	1800225	693000	114345					9305856
6	1	504	24948	406560	3198825	14451696	41011824	76578048	95757795	79571800	42231420	12972960	1756755			367963136
7	1	1016	79389	1896384	21284025	136432296	554475621	1516706304	2887091019	3880894600	3677593535	2407204800	1037071035	264864600	30405375	16416000000

All 15 rows

Technical details

Node id: n-4b763390b67311c8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-76f2941a93f2375f

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,    8,      9;
  2: 1,   24,    108,      160,        75;
  3: 1,   56,    549,     1984,      3275,       2520,         735;
  4: 1,  120,   2160,    14240,     45750,      80136,       78400,        40320,          8505;
  5: 1,  248,   7569,    80960,    424750,    1267776,     2306038,      2610944,       1800225,        693000,        114345;
  6: 1,  504,  24948,   406560,   3198825,   14451696,    41011824,     76578048,      95757795,      79571800,      42231420,      12972960,       1756755;
  7: 1, 1016,  79389,  1896384,  21284025,  136432296,   554475621,   1516706304,    2887091019,    3880894600,    3677593535,    2407204800,    1037071035,     264864600,      30405375;
  8: 1, 2040, 247320,  8440480, 130786500, 1142151192,  6318306960,  23727911040,   63113196870,  121925353000,  173186513016,  181057425120,  137786068420,   74297463240,   26918892000,    5881075200,    585810225;
  9: 1, 4088, 760329, 36416320, 761499500, 8812197072, 64164992556, 317540990208, 1121620808070, 2918367904000, 5704167054806, 8462751979968, 9551777639404, 8152215671600, 5176675003500, 2371510901760, 741049934625, 141421480200, 12439852425;

%F Formula

The row polynomials satisfy P(n,x) = -(6*x^2+12*x^3+6*x^4)*P(n-2,x) - (2*x^2+6*x^3+6*x^4+2*x^5)*d/dx P(n-2,x) + (1+8*x+7*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[-(6*x^2 + 12*x^3 + 6*x^4)*P[n - 2, x] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*D[P[n - 2, x], x] + (1 + 8*x + 7*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*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 + 8*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(6*x^2 + 12*x^3 + 6*x^4)*v[i - 2] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*deriv(v[i - 2], x) + (1 + 8*x + 7*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

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