A088729 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-3\,x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2}\right)\,P_{n-2}''(x) + \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(2 + 3\,x\right)\,P_{n-1}'(x) + \left(2\,x\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 13 + 18\,x + 3\,x^{2}\)

\(P_{2}(x) = 75 + 158\,x + 54\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	13	18	3								34
2	75	158	54	4							291
3	541	1530	795	120	5						2991
4	4683	16622	11610	2660	225	6					35806
5	47293	201978	177303	54600	7000	378	7				488559
6	545835	2724878	2881494	1115604	192150	15708	588	8			7476265
7	7087261	40492890	50101635	23378040	5062155	555660	31458	864	9		126709972
8	102247563	657944942	932458770	509327620	132937875	18341442	1393560	57840	1215	10	2354710837

All 15 rows

Technical details

Node id: n-98e09500466a3809 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c9e06b6396582449

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:          13,           18,            3;
   2:          75,          158,           54,            4;
   3:         541,         1530,          795,          120,           5;
   4:        4683,        16622,        11610,         2660,         225,           6;
   5:       47293,       201978,       177303,        54600,        7000,         378,          7;
   6:      545835,      2724878,      2881494,      1115604,      192150,       15708,        588,         8;
   7:     7087261,     40492890,     50101635,     23378040,     5062155,      555660,      31458,       864,       9;
   8:   102247563,    657944942,    932458770,    509327620,   132937875,    18341442,    1393560,     57840,    1215,     10;
   9:  1622632573,  11611834938,  18546943503,  11609221800,  3547799750,   590007726,   56331429,   3136320,   99495,   1650,   11;
  10: 28091567595, 221291822798, 393306236334, 277569511604, 97228909800, 18933361656, 2178661716, 152385816, 6482025, 162250, 2178, 12;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 3*x^2*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (3+2*x)*P(n-1,x) + (2+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 13 + 18*x + 3*x^2, P(2,x) = 75 + 158*x + 54*x^2 + 4*x^3.

%t Mathematica

P[1, x_] := 13 + 18*x + 3*x^2;
P[2, x_] := 75 + 158*x + 54*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (3 + 2*x)*P[n - 1, x] + (2 + 3*x)*D[P[n - 1, x], x] + 2*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 13 + 18*x + 3*x^2; v[2] = 75 + 158*x + 54*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (3 + 2*x)*v[i - 1] + (2 + 3*x)*deriv(v[i - 1], x) + 2*x*deriv(deriv(v[i - 1], x), 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.