A075503 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-32\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-128\,x + 32\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-64\,x^{2} + 32\,x^{3}\right)\,P_{n-2}''(x) + \left(8 + 4\,x\right)\,P_{n-1}(x) + \left(16\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 64 + 96\,x + 9\,x^{2}\)

\(P_{1}(x) = 512 + 1792\,x + 432\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	64	96	9								169
1	512	1792	432	16							2752
2	4096	30720	14400	1280	25						50521
3	32768	507904	414720	66560	3000	36					1024988
4	262144	8257536	11096064	2867200	224000	6048	49				22713041
5	2097152	133169152	284884992	111476736	13440000	612864	10976	64			545691936
6	16777216	2139095040	7136870400	4073717760	711782400	48771072	1448832	18432	81		14128481233
7	134217728	34292629504	176097853440	143046737920	34836480000	3365978112	147517440	3072000	29160	100	391924515404

All 15 rows

Technical details

Node id: n-9773d624cfbe6ef6 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-cfb8d8d9d8395f5d

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:         64,            96,               9;
  1:        512,          1792,             432,              16;
  2:       4096,         30720,           14400,            1280,             25;
  3:      32768,        507904,          414720,           66560,           3000,             36;
  4:     262144,       8257536,        11096064,         2867200,         224000,           6048,            49;
  5:    2097152,     133169152,       284884992,       111476736,       13440000,         612864,         10976,          64;
  6:   16777216,    2139095040,      7136870400,      4073717760,      711782400,       48771072,       1448832,       18432,        81;
  7:  134217728,   34292629504,    176097853440,    143046737920,    34836480000,     3365978112,     147517440,     3072000,     29160,      100;
  8: 1073741824,  549218942976,   4303506898944,   4890558464000,  1616969728000,   211731480576,   12842446848,   389283840,   5987520,    44000,   121;
  9: 8589934592, 8791798054912, 104519908196352, 164148549779456, 72320286720000, 12491547475968, 1007367094272, 41687973888, 923788800, 10912000, 63888, 144;

%F Formula

The row polynomials satisfy P(n,x) = -(32*x-9*x^2)*P(n-2,x) - (128*x-32*x^2-4*x^3)*d/dx P(n-2,x) - (64*x^2-32*x^3)*d^2/dx^2 P(n-2,x) + (8+4*x)*P(n-1,x) + (16*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 64 + 96*x + 9*x^2, P(1,x) = 512 + 1792*x + 432*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 64 + 96*x + 9*x^2;
P[1, x_] := 512 + 1792*x + 432*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(32*x - 9*x^2)*P[n - 2, x] - (128*x - 32*x^2 - 4*x^3)*D[P[n - 2, x], x] - (64*x^2 - 32*x^3)*D[P[n - 2, x], {x, 2}] + (8 + 4*x)*P[n - 1, x] + (16*x - 4*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] = 64 + 96*x + 9*x^2; v[2] = 512 + 1792*x + 432*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(32*x - 9*x^2)*v[i - 2] - (128*x - 32*x^2 - 4*x^3)*deriv(v[i - 2], x) - (64*x^2 - 32*x^3)*deriv(deriv(v[i - 2], x), x) + (8 + 4*x)*v[i - 1] + (16*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 - 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 Stirling2 triangle with scaled diagonals (powers of 8).?
  - 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.