A322944 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 + 8\,x + 9\,x^{2} + 7\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(12\,x - 8\,x^{2} + 4\,x^{3} - 6\,n x + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(-9\,x^{2} + 12\,x^{3}\right)\,P_{n-2}''(x) + \left(-6 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(6\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	2	24	9													35
4	6	152	135	16												309
5	24	1088	1692	448	25											3277
6	120	8800	21060	9280	1125	36										40421
7	720	79680	272520	176640	34750	2376	49									566735
8	5040	799680	3722040	3310720	936250	102312	4459	64								8880565
9	40320	8816640	53907840	62899200	23898000	3701376	255192	7680	81							153526329
10	362880	105960960	828455040	1228021760	600138000	123246144	11928168	563712	12393	100						2898689157
11	3628800	1378944000	13493692800	24805222400	15121820000	3945352320	501740400	33054720	1134810	19000	121					59284609371
12	39916800	19319731200	232495401600	520093235200	386560020000	124354581120	19920534480	1713085440	81553230	2123000	27951	144				1304580210165
13	479001600	289955635200	4228386624000	11335283097600	10090086160000	3914220119040	766876436480	82550476800	5104040040	183480000	3742772	39744	169			30713128853445
14	6227020800	4641206169600	80991746035200	256893359923200	269963533840000	124127887403520	29100858026240	3808350740480	293327428680	13640484000	382949996	6282432	54925	196		769840526359269
15	87178291200	78925412966400	1630277634585600	6052532604518400	7420447754720000	3988456223431680	1100179303583360	171301559869440	15966386195760	921392472000	33354785464	751016448	10119382	74088	225	20459129566629447

Technical details

Node id: n-be8e1f4bb575779f · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7d7c17087b843f8b

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:      2,        24,         9;
   4:      6,       152,       135,         16;
   5:     24,      1088,      1692,        448,        25;
   6:    120,      8800,     21060,       9280,      1125,        36;
   7:    720,     79680,    272520,     176640,     34750,      2376,       49;
   8:   5040,    799680,   3722040,    3310720,    936250,    102312,     4459,     64;
   9:  40320,   8816640,  53907840,   62899200,  23898000,   3701376,   255192,   7680,    81;
  10: 362880, 105960960, 828455040, 1228021760, 600138000, 123246144, 11928168, 563712, 12393, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-7*n+10+(4*n-8)*x-9*x^2)*P(n-2,x) - ((6*n-12)*x-(4*n-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) + (2*n-6+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) = 1 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 10 + (4*n - 8)*x - 9*x^2)*P[n - 2, x] - ((6*n - 12)*x - (4*n - 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}] + (2*n - 6 + 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] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 7*n + 10 + (4*n - 8)*x - 9*x^2)*v[i - 2] - ((6*n - 12)*x - (4*n - 8)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (9*x^2 - 12*x^3)*deriv(deriv(v[i - 2], x), x) + (2*n - 6 + 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 Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.?
  - 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.