A075501 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-24\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-72\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-36\,x^{2} + 24\,x^{3}\right)\,P_{n-2}''(x) + \left(6 + 4\,x\right)\,P_{n-1}(x) + \left(12\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 36 + 72\,x + 9\,x^{2}\)

\(P_{1}(x) = 216 + 1008\,x + 324\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	36	72	9								117
1	216	1008	324	16							1564
2	1296	12960	8100	960	25						23341
3	7776	160704	174960	37440	2250	36					383166
4	46656	1959552	3510864	1209600	126000	4536	49				6857257
5	279936	23701248	67604544	35271936	5670000	344736	8232	64			132880696
6	1679616	285534720	1270209600	966712320	225212400	20575296	814968	13824	81		2770752825
7	10077696	3433135104	23506225920	25459246080	8266860000	1065016512	62233920	1728000	21870	100	61804545202

All 15 rows

Technical details

Node id: n-8e1f40621980b6fa · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-c952f3936870db53

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:        36,           72,             9;
  1:       216,         1008,           324,             16;
  2:      1296,        12960,          8100,            960,            25;
  3:      7776,       160704,        174960,          37440,          2250,            36;
  4:     46656,      1959552,       3510864,        1209600,        126000,          4536,           49;
  5:    279936,     23701248,      67604544,       35271936,       5670000,        344736,         8232,          64;
  6:   1679616,    285534720,    1270209600,      966712320,     225212400,      20575296,       814968,       13824,        81;
  7:  10077696,   3433135104,   23506225920,    25459246080,    8266860000,    1065016512,     62233920,     1728000,     21870,     100;
  8:  60466176,  41237932032,  430836620544,   652810752000,  287785872000,   50244872832,   4063430448,   164229120,   3367980,   33000,   121;
  9: 362797056, 495097049088, 7847844516864, 16433389817856, 9653592960000, 2223227077632, 239052933504, 13190335488, 389723400, 6138000, 47916, 144;

%F Formula

The row polynomials satisfy P(n,x) = -(24*x-9*x^2)*P(n-2,x) - (72*x-24*x^2-4*x^3)*d/dx P(n-2,x) - (36*x^2-24*x^3)*d^2/dx^2 P(n-2,x) + (6+4*x)*P(n-1,x) + (12*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 36 + 72*x + 9*x^2, P(1,x) = 216 + 1008*x + 324*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 36 + 72*x + 9*x^2;
P[1, x_] := 216 + 1008*x + 324*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(24*x - 9*x^2)*P[n - 2, x] - (72*x - 24*x^2 - 4*x^3)*D[P[n - 2, x], x] - (36*x^2 - 24*x^3)*D[P[n - 2, x], {x, 2}] + (6 + 4*x)*P[n - 1, x] + (12*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] = 36 + 72*x + 9*x^2; v[2] = 216 + 1008*x + 324*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(24*x - 9*x^2)*v[i - 2] - (72*x - 24*x^2 - 4*x^3)*deriv(v[i - 2], x) - (36*x^2 - 24*x^3)*deriv(deriv(v[i - 2], x), x) + (6 + 4*x)*v[i - 1] + (12*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 6).?
  - 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.