A166973 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 - 4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-10\,x + 4\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-25\,x^{2} + 20\,x^{3}\right)\,P_{n-2}''(x) + \left(-3 + 4\,x\right)\,P_{n-1}(x) + \left(10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	1	28	9													38
4	1	172	162	16												351
5	1	1036	2169	544	25											3775
6	1	6220	26190	12560	1375	36										46382
7	1	37324	302085	247520	48500	2916	49									638395
8	1	223948	3406914	4497360	1405250	145656	5488	64								9684681
9	1	1343692	37979937	78014496	36537375	5810616	368382	9472	81							160064052
10	1	8062156	420802614	1315751824	889182525	203689836	19328736	822144	15309	100						2857655245
11	1	48372940	4646968605	21800122720	20728695250	6576358572	876435315	54842880	1668195	23500	121					54733488099
12	1	290237644	51225493770	357063241040	469365292000	200834644032	36120649488	3119075520	137806515	3140500	34606	144				1118159615260
13	1	1741425868	564133466169	5804079401120	10414582446125	5897586765312	1393098399616	159464709888	9597647070	314594500	5564911	49248	169			24244604469997
14	1	10448555212	6209386336062	93868174357776	227775105432875	168334254620532	51213321263104	7560286649344	595326053322	26320294000	664469806	9380592	68107	196		555593297480929
15	1	62691331276	68326758945909	1512929698766304	4929946236524400	4704686771957172	1816734583500837	339061187964928	33976855976778	1945703960200	65735515846	1316087136	15163694	91924	225	13407737555786630

Technical details

Node id: n-b6eaf2e92e68b0d0 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ac8449946d0042ca

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: 1,       4;
   3: 1,      28,         9;
   4: 1,     172,       162,         16;
   5: 1,    1036,      2169,        544,        25;
   6: 1,    6220,     26190,      12560,      1375,        36;
   7: 1,   37324,    302085,     247520,     48500,      2916,       49;
   8: 1,  223948,   3406914,    4497360,   1405250,    145656,     5488,     64;
   9: 1, 1343692,  37979937,   78014496,  36537375,   5810616,   368382,   9472,    81;
  10: 1, 8062156, 420802614, 1315751824, 889182525, 203689836, 19328736, 822144, 15309, 100;

%F Formula

The row polynomials satisfy P(n,x) = (4-4*x+9*x^2)*P(n-2,x) - (10*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (25*x^2-20*x^3)*d^2/dx^2 P(n-2,x) - (3-4*x)*P(n-1,x) + (10*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(4 - 4*x + 9*x^2)*P[n - 2, x] - (10*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (25*x^2 - 20*x^3)*D[P[n - 2, x], {x, 2}] - (3 - 4*x)*P[n - 1, x] + (10*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] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = (4 - 4*x + 9*x^2)*v[i - 2] - (10*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (25*x^2 - 20*x^3)*deriv(deriv(v[i - 2], x), x) - (3 - 4*x)*v[i - 1] + (10*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 T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m…?
  - 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.