A166979 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 - 4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-18\,x + 4\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-81\,x^{2} + 36\,x^{3}\right)\,P_{n-2}''(x) + \left(-7 + 4\,x\right)\,P_{n-1}(x) + \left(18\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1 + 44\,x + 9\,x^{2}\)

\(P_{2}(x) = 1 + 444\,x + 270\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	1	44	9								54
2	1	444	270	16							731
3	1	4444	6129	928	25						11527
4	1	44444	126450	36880	2375	36					210186
5	1	444444	2502549	1257440	145500	5076	49				4355059
6	1	4444444	48548430	39657296	7348250	443016	9604	64			100451105
7	1	44444444	932420169	1196712608	333849775	30960216	1131214	16640	81		2539535148
8	1	444444444	17815983210	35165588880	14222305125	1904913612	104357064	2542464	26973	100	69660161873

All 15 rows

Technical details

Node id: n-bfbd37281ac07ff8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3b0795cfa52da853

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 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,          44,             9;
   2: 1,         444,           270,             16;
   3: 1,        4444,          6129,            928,             25;
   4: 1,       44444,        126450,          36880,           2375,            36;
   5: 1,      444444,       2502549,        1257440,         145500,          5076,           49;
   6: 1,     4444444,      48548430,       39657296,        7348250,        443016,         9604,          64;
   7: 1,    44444444,     932420169,     1196712608,      333849775,      30960216,      1131214,       16640,        81;
   8: 1,   444444444,   17815983210,    35165588880,    14222305125,    1904913612,    104357064,     2542464,     26973,     100;
   9: 1,  4444444444,  339503680989,  1016309347680,   581171522250,  108106145532,   8332437603,   299020800,   5186835,   41500,   121;
  10: 1, 44444444444, 6460569938790, 29060223834576, 23091329679000, 5809769686512, 605428544028, 30020515008, 757087155, 9806500, 61226, 144;

%F Formula

The row polynomials satisfy P(n,x) = (8-4*x+9*x^2)*P(n-2,x) - (18*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (81*x^2-36*x^3)*d^2/dx^2 P(n-2,x) - (7-4*x)*P(n-1,x) + (18*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 44*x + 9*x^2, P(2,x) = 1 + 444*x + 270*x^2 + 16*x^3.

%t Mathematica

P[1, x_] := 1 + 44*x + 9*x^2;
P[2, x_] := 1 + 444*x + 270*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[(8 - 4*x + 9*x^2)*P[n - 2, x] - (18*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (81*x^2 - 36*x^3)*D[P[n - 2, x], {x, 2}] - (7 - 4*x)*P[n - 1, x] + (18*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 44*x + 9*x^2; v[2] = 1 + 444*x + 270*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (8 - 4*x + 9*x^2)*v[i - 2] - (18*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (81*x^2 - 36*x^3)*deriv(deriv(v[i - 2], x), x) - (7 - 4*x)*v[i - 1] + (18*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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 Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m…?
  - 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.