A084950 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9\,x^{2}\right)\,P_{n-4}(x) + \left(4\,x^{3}\right)\,P_{n-4}'(x) + \left(4\,x - 4\,n x\right)\,P_{n-3}(x) + \left(-4\,x^{2} + 4\,n x^{2}\right)\,P_{n-3}'(x) + \left(4\,x + n - n^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2}\right)\,P_{n-2}'(x) + \left(2\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 6 + 16\,x\)

\(P_{3}(x) = 24 + 72\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	6	16							22
3	24	72	9						105
4	120	384	81						585
5	720	2400	648	16					3784
6	5040	17280	5400	256					27976
7	40320	141120	48600	3200	25				233265
8	362880	1290240	476280	38400	625				2168425
9	3628800	13063680	5080320	470400	11250	36			22254486
10	39916800	145152000	58786560	6021120	183750	1296			250061526
11	479001600	1756339200	734832000	81285120	2940000	31752	49		3054429721
12	6227020800	22992076800	9879408000	1161216000	47628000	677376	2401		40308029377
13	87178291200	323805081600	142263475200	17563392000	793800000	13716864	76832	64	571617833760
14	1307674368000	4881984307200	2185684300800	281014272000	13721400000	274337280	2074464	4096	8670355063840

Technical details

Node id: n-926cc35d1e5131de · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-83af54e57a7f6434

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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:       2,        4;
  2:       6,       16;
  3:      24,       72,       9;
  4:     120,      384,      81;
  5:     720,     2400,     648,     16;
  6:    5040,    17280,    5400,    256;
  7:   40320,   141120,   48600,   3200,    25;
  8:  362880,  1290240,  476280,  38400,   625;
  9: 3628800, 13063680, 5080320, 470400, 11250, 36;

%F Formula

The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - (4*n-4)*x*P(n-3,x) + (4*n-4)*x^2*d/dx P(n-3,x) - (n^2-n-4*x)*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + 2*n*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 16*x, P(3,x) = 24 + 72*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 6 + 16*x;
P[3, x_] := 24 + 72*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - (4*n - 4)*x*P[n - 3, x] + (4*n - 4)*x^2*D[P[n - 3, x], x] - (n^2 - n - 4*x)*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + 2*n*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] = 2 + 4*x; v[3] = 6 + 16*x; v[4] = 24 + 72*x + 9*x^2; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - (4*n - 4)*x*v[i - 3] + (4*n - 4)*x^2*deriv(v[i - 3], x) - (n^2 - n - 4*x)*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + 2*n*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

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 Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+...…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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.