A071951 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	2														4
3	4	16	3													23
4	8	104	60	4												176
5	16	640	876	160	5											1697
6	32	3872	11472	4368	350	6										20100
7	64	23296	143472	102656	15960	672	7									286127
8	128	139904	1756608	2244416	607120	47376	1176	8								4796736
9	256	839680	21289152	47230464	21019120	2718336	121128	1920	9							93220065
10	512	5038592	256729344	972994816	689611680	139393056	9954560	276672	2970	10						2074002212
11	1024	30232576	3088310016	19802202112	21904593920	6682042368	720080592	31297024	578556	4400	11					52259342599
12	2048	181397504	37105069056	400161788928	681890570240	306931292160	48120229248	3076334976	87279192	1126840	6292	12				1477555096496
13	4096	1088389120	445532924928	8052709203968	20956919343360	13709382955008	3052819345408	276490665984	11316004128	220929280	2070068	8736	13			46506481844097
14	8192	6530342912	5348027682816	161648227979264	638773466805760	600942387322368	186952163457024	23396264345600	1329492370752	36875558720	516271184	3621072	11830	14		1618433955777508
15	16384	39182073856	64186127708160	3240095263162368	19365264289146880	26006108427706368	11170420605469440	1898190648262656	145975110756480	5533525204480	108710910880	1128092160	6075888	15680	15	61895923024601695

Technical details

Node id: n-688020112ddc2e19 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-dbde9fb02b4eeb4a

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 -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;
   2:   2,       2;
   3:   4,      16,         3;
   4:   8,     104,        60,         4;
   5:  16,     640,       876,       160,         5;
   6:  32,    3872,     11472,      4368,       350,         6;
   7:  64,   23296,    143472,    102656,     15960,       672,       7;
   8: 128,  139904,   1756608,   2244416,    607120,     47376,    1176,      8;
   9: 256,  839680,  21289152,  47230464,  21019120,   2718336,  121128,   1920,    9;
  10: 512, 5038592, 256729344, 972994816, 689611680, 139393056, 9954560, 276672, 2970, 10;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (2 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*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] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (2 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + 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, 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 Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.?
  - 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.