A225468 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	4														6
3	4	28	9													41
4	8	156	135	16												315
5	16	812	1431	416	25											2700
6	32	4124	13275	7120	1000	36										25587
7	64	20748	115479	101920	25125	2052	49									265437
8	128	103996	970515	1326416	511000	71064	3773	64								2986956
9	256	520492	7998111	16315936	9226525	1943928	172186	6400	81							36183915
10	512	2603484	65155995	193694160	154665000	46332972	6089622	372096	10206	100						468924147
11	1024	13019468	527105799	2246468640	2467957125	1010378124	184856763	16512000	736290	15500	121					6467050854
12	2048	65101436	4246140195	25648232016	38061507000	20730286368	5072372151	621221568	40041540	1358500	22627	144				94486285593
13	4096	325515372	34115599791	289679245856	572936460525	407223438336	129663666132	20913235200	1827313587	88830500	2367849	31968	169			1456775709381
14	8192	1627593244	273657207915	3247121659600	8473734269000	7747826954868	3147549669264	650361034752	73978466562	4832027200	183256073	3936816	43940	196		23620876127622
15	16384	8137998988	2192919748119	36204839958560	123705907359125	143915235580116	73496646740517	19069389081600	2746553315220	231460229000	11710947248	355878432	6290011	58996	225	401583163202541

Technical details

Node id: n-d8ea3bd27d0a5d8c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-b7d26c421500f1e6

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:   2,       4;
   3:   4,      28,        9;
   4:   8,     156,      135,        16;
   5:  16,     812,     1431,       416,        25;
   6:  32,    4124,    13275,      7120,      1000,       36;
   7:  64,   20748,   115479,    101920,     25125,     2052,      49;
   8: 128,  103996,   970515,   1326416,    511000,    71064,    3773,     64;
   9: 256,  520492,  7998111,  16315936,   9226525,  1943928,  172186,   6400,    81;
  10: 512, 2603484, 65155995, 193694160, 154665000, 46332972, 6089622, 372096, 10206, 100;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 - 8*x + 9*x^2)*P[n - 2, x] - (12*x - 8*x^2 - 4*x^3)*D[P[n - 2, x], x] - (9*x^2 - 12*x^3)*D[P[n - 2, x], {x, 2}] + (1 + 4*x)*P[n - 1, x] + (6*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] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (2 - 8*x + 9*x^2)*v[i - 2] - (12*x - 8*x^2 - 4*x^3)*deriv(v[i - 2], x) - (9*x^2 - 12*x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 4*x)*v[i - 1] + (6*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, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 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.