A285061 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 - 16\,x + 144\,x^{2}\right)\,P_{n-2}(x) + \left(-8\,x + 16\,x^{2} + 64\,x^{3}\right)\,P_{n-2}'(x) + \left(-16\,x^{2} + 64\,x^{3}\right)\,P_{n-2}''(x) + \left(-2 + 16\,x\right)\,P_{n-1}(x) + \left(8\,x - 16\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 16\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	16														17
2	1	96	144													241
3	1	496	2160	1024												3681
4	1	2496	23904	28672	6400											61473
5	1	12496	237600	542720	288000	36864										1117681
6	1	62496	2250864	8744960	8288000	2433024	200704									21980049
7	1	312496	20820240	129690624	195552000	98832384	18264064	1048576								464520385
8	1	1562496	190194624	1834033152	4134950400	3201859584	994689024	125829120	5308416							10488426817
9	1	7812496	1725814080	25194926080	81756864000	91056365568	42299572224	8845787136	812187648	26214400						251725543633
10	1	39062496	15602639184	339806494720	1547334976000	2383103213568	1553240629248	477521510400	71583989760	4980736000	126877696					6393340129073
11	1	195312496	140775315120	4528436532224	28428485184000	58957816946688	51805688782848	21963013619712	4779724308480	537788416000	29308747776	603979776				171171837145121
12	1	976562496	1268735648544	59870743826432	511586976454400	1401862230540288	1616134778503168	907585687388160	268918658629632	43501748224000	3804554592256	166698418176	2835349504			4814704624137057
13	1	4882812496	11427409899360	787341789911040	9071170748640000	32385847825723392	48035730495520768	34763464593375232	13468968277180416	2937558060236800	366535199686656	25612366381056	921488588800	13153337344		141854596291293361
14	1	24414062496	102890634406704	10316704850350080	159130788913824000	732352747852357632	1377216211661402112	1259102657021214720	620465992650915840	175203071819776000	29245724198699008	2897385206317056	165388771917824	4971961516032	60397977600	4366204620354737105

Technical details

Node id: n-e6ad03919fa0e1c6 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3e4519a82a1d1463

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 0)
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: 1,      16;
  2: 1,      96,        144;
  3: 1,     496,       2160,        1024;
  4: 1,    2496,      23904,       28672,        6400;
  5: 1,   12496,     237600,      542720,      288000,       36864;
  6: 1,   62496,    2250864,     8744960,     8288000,     2433024,      200704;
  7: 1,  312496,   20820240,   129690624,   195552000,    98832384,    18264064,    1048576;
  8: 1, 1562496,  190194624,  1834033152,  4134950400,  3201859584,   994689024,  125829120,   5308416;
  9: 1, 7812496, 1725814080, 25194926080, 81756864000, 91056365568, 42299572224, 8845787136, 812187648, 26214400;

%F Formula

The row polynomials satisfy P(n,x) = (3-16*x+144*x^2)*P(n-2,x) - (8*x-16*x^2-64*x^3)*d/dx P(n-2,x) - (16*x^2-64*x^3)*d^2/dx^2 P(n-2,x) - (2-16*x)*P(n-1,x) + (8*x-16*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 16*x;
P[n_, x_] := P[n, x] = Expand[(3 - 16*x + 144*x^2)*P[n - 2, x] - (8*x - 16*x^2 - 64*x^3)*D[P[n - 2, x], x] - (16*x^2 - 64*x^3)*D[P[n - 2, x], {x, 2}] - (2 - 16*x)*P[n - 1, x] + (8*x - 16*x^2)*D[P[n - 1, x], 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] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (3 - 16*x + 144*x^2)*v[i - 2] - (8*x - 16*x^2 - 64*x^3)*deriv(v[i - 2], x) - (16*x^2 - 64*x^3)*deriv(deriv(v[i - 2], x), x) - (2 - 16*x)*v[i - 1] + (8*x - 16*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 - 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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 Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).?
  - 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.