A075500 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-50\,x + 20\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-25\,x^{2} + 20\,x^{3}\right)\,P_{n-2}''(x) + \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 25 + 60\,x + 9\,x^{2}\)

\(P_{1}(x) = 125 + 700\,x + 270\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	25	60	9								94
1	125	700	270	16							1111
2	625	7500	5625	800	25						14575
3	3125	77500	101250	26000	1875	36					209786
4	15625	787500	1693125	700000	87500	3780	49				3287579
5	78125	7937500	27168750	17010000	3281250	239400	6860	64			55721949
6	390625	79687500	425390625	388500000	108609375	11907000	565950	11520	81		1015062676
7	1953125	798437500	6560156250	8526250000	3322265625	513607500	36015000	1200000	18225	100	19759903325

All 15 rows

Technical details

Node id: n-5d54cd7ef0177801 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-1cc6c3f42efa30d8

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:       25,          60,             9;
  1:      125,         700,           270,            16;
  2:      625,        7500,          5625,           800,            25;
  3:     3125,       77500,        101250,         26000,          1875,           36;
  4:    15625,      787500,       1693125,        700000,         87500,         3780,          49;
  5:    78125,     7937500,      27168750,      17010000,       3281250,       239400,        6860,         64;
  6:   390625,    79687500,     425390625,     388500000,     108609375,     11907000,      565950,      11520,        81;
  7:  1953125,   798437500,    6560156250,    8526250000,    3322265625,    513607500,    36015000,    1200000,     18225,     100;
  8:  9765625,  7992187500,  100198828125,  182187500000,   96378906250,  20192287500,  1959601875,   95040000,   2338875,   27500,   121;
  9: 48828125, 79960937500, 1520964843750, 3821881250000, 2694140625000, 744554250000, 96070012500, 6361080000, 225534375, 4262500, 39930, 144;

%F Formula

The row polynomials satisfy P(n,x) = -(20*x-9*x^2)*P(n-2,x) - (50*x-20*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) + (5+4*x)*P(n-1,x) + (10*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 25 + 60*x + 9*x^2, P(1,x) = 125 + 700*x + 270*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 25 + 60*x + 9*x^2;
P[1, x_] := 125 + 700*x + 270*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(20*x - 9*x^2)*P[n - 2, x] - (50*x - 20*x^2 - 4*x^3)*D[P[n - 2, x], x] - (25*x^2 - 20*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 4*x)*P[n - 1, x] + (10*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] = 25 + 60*x + 9*x^2; v[2] = 125 + 700*x + 270*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(20*x - 9*x^2)*v[i - 2] - (50*x - 20*x^2 - 4*x^3)*deriv(v[i - 2], x) - (25*x^2 - 20*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 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 - 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 5).?
  - 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.