A035469 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-30 + 20\,x + 9\,x^{2} + 33\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(10\,x - 20\,x^{2} + 4\,x^{3} - 6\,n x + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(-8 + 4\,x + 6\,n\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	4														8
3	28	48	9													85
4	280	640	216	16												1152
5	3640	10080	4680	640	25											19065
6	58240	185920	106920	20480	1500	36										373096
7	1106560	3951360	2663640	640640	66500	3024	49									8431773
8	24344320	95307520	72817920	20751360	2730000	177408	5488	64								216134080
9	608608000	2575372800	2180525760	710492160	111594000	9253440	411600	9216	81							6196267057
10	17041024000	77120243200	71210361600	25901747200	4681152000	466058880	26589360	860160	14580	100						196448051080
11	528271744000	2536011878400	2523462480000	1007255603200	204311800000	23518978560	1618164240	67415040	1657260	22000	121					6824519742821
12	17961239296000	90873502720000	96550676006400	41754612838400	9337685280000	1211449155840	96738512640	4877537280	154927080	2992000	31944	144				257790939297728
13	664565853952000	3525038060544000	3969941745369600	1841830159769600	448086679040000	64327131348480	5808672950080	340963983360	13144851720	328900000	5121688	44928	169			10519952745875625
14	26582634158080000	147184823898112000	174668888941747200	86256371084083200	22593673502400000	3539965728499200	354755328928000	23612145356800	1062485424000	32344312000	654053400	8386560	61516	196		461205820279444072
15	1143053268797440000	6582462788149248000	8191265856149376000	4278315539097600000	1196678234431680000	202453244811417600	22201297803484800	1643961166848000	84070878091200	2993614624000	73801447720	1231251840	13225940	82320	225	21618161329945817645

Technical details

Node id: n-9a51e72a74f2d3e0 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-b01a90078e4aa5ae

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:           1;
   2:           4,           4;
   3:          28,          48,           9;
   4:         280,         640,         216,          16;
   5:        3640,       10080,        4680,         640,         25;
   6:       58240,      185920,      106920,       20480,       1500,        36;
   7:     1106560,     3951360,     2663640,      640640,      66500,      3024,       49;
   8:    24344320,    95307520,    72817920,    20751360,    2730000,    177408,     5488,     64;
   9:   608608000,  2575372800,  2180525760,   710492160,  111594000,   9253440,   411600,   9216,    81;
  10: 17041024000, 77120243200, 71210361600, 25901747200, 4681152000, 466058880, 26589360, 860160, 14580, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-33*n+30+(12*n-20)*x-9*x^2)*P(n-2,x) - ((6*n-10)*x-(12*n-20)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (6*n-8+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 33*n + 30 + (12*n - 20)*x - 9*x^2)*P[n - 2, x] - ((6*n - 10)*x - (12*n - 20)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (6*n - 8 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 4*x; for(n = 3, N, my(i = n); v[i] = -(9*n^2 - 33*n + 30 + (12*n - 20)*x - 9*x^2)*v[i - 2] - ((6*n - 10)*x - (12*n - 20)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (6*n - 8 + 4*x)*v[i - 1] + (2*x - 4*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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 read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
[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.