A094262 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 8\,x - 12\,x^{2} - 8\,x^{3} - 2\,x^{4}\right)\,P_{n-2}(x) + \left(-x - 4\,x^{2} - 6\,x^{3} - 4\,x^{4} - x^{5}\right)\,P_{n-2}'(x) + \left(3 + 8\,x + 5\,x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	P_n(1)
0																													0
1	2	2																											4
2	6	24	30	12																									72
3	14	122	372	524	350	90																							1472
4	30	480	2670	7320	11130	9600	4410	840																					36480
5	62	1682	15180	67960	176080	282372	285572	177800	62370	9450																			1078528
6	126	5544	76230	511812	2007180	5021856	8375724	9457560	7161462	3490200	990990	124740																	37223424
7	254	17642	355572	3405444	18948930	67894974	165889752	285144792	349280514	303933350	183883700	73638180	17567550	1891890															1471852544
8	510	54960	1582590	20925840	158632110	773670240	2595240270	6233402160	10973281770	14312934960	13830775530	9783626160	4927892970	1674953280	344594250	32432400													65664000000
9	1022	168962	6828060	121839920	1223916260	7856937864	34731045804	110777363760	262653111360	471418764860	646460220692	678232731792	540708182300	322104222440	138955716900	41026986000	7420262850	620269650											3264318570496
10	2046	515064	28839030	683046012	8912016960	73536758592	416459248128	1705409674320	5227006343724	12268587563760	22376061452316	31965257169384	35841451142640	31422524445312	21311158268400	10962267235920	4135394933670	1079269191000	174157933950	13094581500									178981260361728
11	4094	1561562	120003972	3725936764	62202437110	648213199458	4607635317824	23650284971984	91088190180108	270296018564660	629298189560744	1163339671613352	1719468294743212	2036901870767764	1929965534296800	1452451259539280	856891349303990	387791000046450	129968260842420	30392523661500	4427278005150	302484832650							10731554109390848
12	8190	4717440	494094510	19919095560	420588858690	5461152016320	47971065272130	303156457563960	1437192992895660	5263503914148480	15206042026195020	35167493957253840	65764549383345540	100046095507088640	124127320694553540	125486146100755440	102890554393576950	67808841360099840	35392730840657190	14297289711705000	4310693017681050	913100881492800	121223088236250	7589619437400					698597397170749440
13	16382	14217842	2018884140	104894123880	2775815810040	44433936896076	476500284856596	3656564063775480	21004264865997270	93246675267575830	327414991172124856	924777500705984016	2126904094443404640	4017119195440432200	6264154899813430440	8085219619732265712	8636059088902079562	7610236368079895550	5497658445900480860	3221983314136987400	1507711351784437800	549971133620208300	150688380216874500	29171967244219800	3557634111281250	205552193096250			49071264858659356672
14	32766	42784584	8203595430	546090690612	17979530636340	351796011866208	4561073434293012	42036941384291880	288989216738522586	1533611215127414280	6442268228347466658	21825283391579971212	60467896800230894520	138416918873479134336	263705848347510150840	420130121275678161456	561117631213015736274	628410568550812159320	588945394189558113162	459802550762092437900	296786300591551510740	156575464369226142240	66393796914897952500	22076097245648245800	5542873003911993750	988020874815975000	111425100365328750	5976825306952500	3699614005334592454656

Technical details

Node id: n-3a888a6c3242a89e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8c48c943a9ebc567

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    0;
  1:    2,      2;
  2:    6,     24,      30,        12;
  3:   14,    122,     372,       524,        350,         90;
  4:   30,    480,    2670,      7320,      11130,       9600,        4410,          840;
  5:   62,   1682,   15180,     67960,     176080,     282372,      285572,       177800,        62370,         9450;
  6:  126,   5544,   76230,    511812,    2007180,    5021856,     8375724,      9457560,      7161462,      3490200,       990990,       124740;
  7:  254,  17642,  355572,   3405444,   18948930,   67894974,   165889752,    285144792,    349280514,    303933350,    183883700,     73638180,     17567550,      1891890;
  8:  510,  54960, 1582590,  20925840,  158632110,  773670240,  2595240270,   6233402160,  10973281770,  14312934960,  13830775530,   9783626160,   4927892970,   1674953280,    344594250,    32432400;
  9: 1022, 168962, 6828060, 121839920, 1223916260, 7856937864, 34731045804, 110777363760, 262653111360, 471418764860, 646460220692, 678232731792, 540708182300, 322104222440, 138955716900, 41026986000, 7420262850, 620269650;

%F Formula

The row polynomials satisfy P(n,x) = -(2+8*x+12*x^2+8*x^3+2*x^4)*P(n-2,x) - (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-2,x) + (3+8*x+5*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 8*x + 12*x^2 + 8*x^3 + 2*x^4)*P[n - 2, x] - (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 2, x], x] + (3 + 8*x + 5*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*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] = 0; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 8*x + 12*x^2 + 8*x^3 + 2*x^4)*v[i - 2] - (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 2], x) + (3 + 8*x + 5*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - 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.