A075505 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-40\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-200\,x + 40\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-100\,x^{2} + 40\,x^{3}\right)\,P_{n-2}''(x) + \left(10 + 4\,x\right)\,P_{n-1}(x) + \left(20\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 100 + 120\,x + 9\,x^{2}\)

\(P_{1}(x) = 1000 + 2800\,x + 540\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	100	120	9								229
1	1000	2800	540	16							4356
2	10000	60000	22500	1600	25						94125
3	100000	1240000	810000	104000	3750	36					2257786
4	1000000	25200000	27090000	5600000	350000	7560	49				59247609
5	10000000	508000000	869400000	272160000	26250000	957600	13720	64			1686781384
6	100000000	10200000000	27225000000	12432000000	1737750000	95256000	2263800	23040	81		51792292921
7	1000000000	204400000000	839700000000	545680000000	106312500000	8217720000	288120000	4800000	36450	100	1705603176550

All 15 rows

Technical details

Node id: n-65ef9e4a573f76ee · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-eeb15c314e2d2fa8

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:          100,            120,               9;
  1:         1000,           2800,             540,              16;
  2:        10000,          60000,           22500,            1600,              25;
  3:       100000,        1240000,          810000,          104000,            3750,             36;
  4:      1000000,       25200000,        27090000,         5600000,          350000,           7560,            49;
  5:     10000000,      508000000,       869400000,       272160000,        26250000,         957600,         13720,           64;
  6:    100000000,    10200000000,     27225000000,     12432000000,      1737750000,       95256000,       2263800,        23040,         81;
  7:   1000000000,   204400000000,    839700000000,    545680000000,    106312500000,     8217720000,     288120000,      4800000,      36450,      100;
  8:  10000000000,  4092000000000,  25650900000000,  23320000000000,   6168250000000,   646153200000,   31353630000,    760320000,    9355500,    55000,   121;
  9: 100000000000, 81880000000000, 778734000000000, 978401600000000, 344850000000000, 47651472000000, 3074240400000, 101777280000, 1804275000, 17050000, 79860, 144;

%F Formula

The row polynomials satisfy P(n,x) = -(40*x-9*x^2)*P(n-2,x) - (200*x-40*x^2-4*x^3)*d/dx P(n-2,x) - (100*x^2-40*x^3)*d^2/dx^2 P(n-2,x) + (10+4*x)*P(n-1,x) + (20*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 100 + 120*x + 9*x^2, P(1,x) = 1000 + 2800*x + 540*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 100 + 120*x + 9*x^2;
P[1, x_] := 1000 + 2800*x + 540*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(40*x - 9*x^2)*P[n - 2, x] - (200*x - 40*x^2 - 4*x^3)*D[P[n - 2, x], x] - (100*x^2 - 40*x^3)*D[P[n - 2, x], {x, 2}] + (10 + 4*x)*P[n - 1, x] + (20*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] = 100 + 120*x + 9*x^2; v[2] = 1000 + 2800*x + 540*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(40*x - 9*x^2)*v[i - 2] - (200*x - 40*x^2 - 4*x^3)*deriv(v[i - 2], x) - (100*x^2 - 40*x^3)*deriv(deriv(v[i - 2], x), x) + (10 + 4*x)*v[i - 1] + (20*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 10).?
  - 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.