A160562 → θ+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	20	3													24
3	1	182	105	4												292
4	1	1640	2898	336	5											4880
5	1	14762	74910	20328	825	6										110832
6	1	132860	1894893	1095952	92235	1716	7									3217664
7	1	1195742	47571615	56228172	8840975	318318	3185	8								114158016
8	1	10761680	1191083988	2818609248	786404190	49125648	909636	5440	9							4856899840
9	1	96855122	29793242220	139699965136	67222000950	6887888436	211041740	2263584	8721	10						243913265920
10	1	871696100	744976338183	6885022614624	5619607033370	914100901896	43701923902	750496960	5066901	13300	11					14209036085248
11	1	7845264902	18625715998725	338359409900820	463794447971250	117349737569460	8452076191650	218806871888	2308643469	10431190	19481	12				946810358862848
12	1	70607384120	465654667865478	16604445373138480	37990299548047275	14755871583470160	1565308903553220	58891061822400	913355693415	6330819000	20065430	27600	13			71441461451886592
13	1	635466457082	11641472607713130	814238696174272824	3097969820108252375	1831048821057546090	281752388214542700	15039413371243680	330212239947135	3300265318350	15812755530	36489960	38025	14		6052024775354576896
14	1	5719198113740	291037768392513873	39913218076016319216	251953353798986283405	225274471132092979740	49752377232824853405	3705870737917876800	112350677387371155	1558298268754500	10603717038915	36553467600	63296415	51156	15	571004254081418919936

Technical details

Node id: n-9718b3717ad8e9e3 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-98f5f68794f5eec6

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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: 1;
  1: 1,        2;
  2: 1,       20,           3;
  3: 1,      182,         105,            4;
  4: 1,     1640,        2898,          336,           5;
  5: 1,    14762,       74910,        20328,         825,          6;
  6: 1,   132860,     1894893,      1095952,       92235,       1716,         7;
  7: 1,  1195742,    47571615,     56228172,     8840975,     318318,      3185,       8;
  8: 1, 10761680,  1191083988,   2818609248,   786404190,   49125648,    909636,    5440,    9;
  9: 1, 96855122, 29793242220, 139699965136, 67222000950, 6887888436, 211041740, 2263584, 8721, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-2,x) - (8*x+4*x^2)*d/dx P(n-2,x) - (4*x^2+4*x^3)*d^2/dx^2 P(n-2,x) + (2+2*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 2, x] - (8*x + 4*x^2)*D[P[n - 2, x], x] - (4*x^2 + 4*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 2*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 2] - (8*x + 4*x^2)*deriv(v[i - 2], x) - (4*x^2 + 4*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 2*x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*x^2*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 - 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of scaled central factorial numbers, T(n,k) = A008958(n,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.