A094262 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 7\,x - 8\,x^{2} - 2\,x^{3} + 2\,x^{4} + x^{5}\right)\,P_{n-2}(x) + \left(-x - 3\,x^{2} - 2\,x^{3} + 2\,x^{4} + 3\,x^{5} + x^{6}\right)\,P_{n-2}'(x) + \left(3 + 7\,x + 3\,x^{2} - x^{3}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 4\,x + x^{2}\)

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	x²⁸	P_n(1)
0	1																													1
1	3	4	1																											8
2	7	30	42	22	3																									104
3	15	136	433	648	481	160	15																							1888
4	31	510	2910	8210	12960	11826	6010	1470	105																					44032
5	63	1744	16021	73020	193070	317588	332634	218596	84595	16380	945																			1254656
6	127	5670	79002	537222	2135133	5423292	9212700	10654092	8343657	4285918	1340010	214830	10395																	42242048
7	255	17896	364393	3523968	19800291	71684760	177205581	308843328	384923613	342742296	214277035	90354880	23704065	3243240	135135															1640820736
8	511	55470	1610070	21453370	163863570	805396662	2724185310	6604150770	11752457040	15532188490	15262069026	11040969390	5743195150	2054021970	464233770	55405350	2027025													72227282944
9	1023	169984	6912541	124115940	1254376240	8101721116	36040535448	115738941732	276500281830	500602443900	693602097178	737001842764	597227576616	363697159540	161963161360	50290700460	9984449475	1056755700	34459425											3553227702272
10	2047	517110	29096562	692659022	9082778463	75319161984	428715374560	1764903852624	5440182553014	12849366046396	23602920208692	33999444574140	38505222573422	34179559148592	23555624300208	12383011120480	4820536635915	1322527716510	234117333450	22260788550	654729075									193194171170816
11	4095	1565656	120784753	3765938088	63133921301	660653686880	4715670851067	24308518588816	94044475801606	280416928584672	656327791417210	1220548597937056	1816413267377658	2169168662671088	2075458525065926	1581115628492400	947669553025195	438196373534920	151512205289445	37232958442680	5946904188225	513307594800	13749310575							11504330794074112
12	8191	4725630	496453230	20083793730	425568632580	5545269788058	48881257274850	310009466888550	1475087550091155	5423192024470220	15732392417609868	36549861414180660	68695173879783360	105104906998115220	131273470373631300	133761300813725676	110733438524874165	73861226912663190	39159888693996070	16160065019108010	5025557503266300	1118371977572850	162727673758650	12860188491150	316234143225					744614720347045888
13	16383	14234224	2025993061	105567085260	2802039341010	44989100058084	483905941005942	3724635533040708	21461335373969205	95580482474908860	336739658698882439	954542499903449912	2203968886168903308	4180727202705309480	6551091985202032740	8502829946386494408	9141385315135346169	8118239843897664936	5920449355238252835	3511333758658065340	1668810517491287170	621766912276610100	175687068108702150	35723635949301300	4773132746457075	347857557547500	7905853580625			52049723740047540224
14	32767	42817350	8224987722	548825222422	18116053308993	355391917993476	4619706102937380	42688523303476596	294243834411559071	1565721128098361234	6595629349860208086	22410944139611559090	62286670416195892121	143068295550419972376	273592771124187231864	437710511165512171512	587375763792745621365	661417488033930732042	623857092442381010902	490799676772069180698	319776428129656132635	170708145349776214180	73510863477135504420	24962784068035113300	6462710389147337325	1209735794972454750	149425903242866250	10103680876038750	213458046676875	3907812900294782615552

Technical details

Node id: n-3d2f72d29c0e6cb2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-75519f96f3248de5

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:    3,      4,       1;
  2:    7,     30,      42,        22,          3;
  3:   15,    136,     433,       648,        481,        160,          15;
  4:   31,    510,    2910,      8210,      12960,      11826,        6010,         1470,          105;
  5:   63,   1744,   16021,     73020,     193070,     317588,      332634,       218596,        84595,        16380,          945;
  6:  127,   5670,   79002,    537222,    2135133,    5423292,     9212700,     10654092,      8343657,      4285918,      1340010,       214830,        10395;
  7:  255,  17896,  364393,   3523968,   19800291,   71684760,   177205581,    308843328,    384923613,    342742296,    214277035,     90354880,     23704065,      3243240,       135135;
  8:  511,  55470, 1610070,  21453370,  163863570,  805396662,  2724185310,   6604150770,  11752457040,  15532188490,  15262069026,  11040969390,   5743195150,   2054021970,    464233770,    55405350,    2027025;
  9: 1023, 169984, 6912541, 124115940, 1254376240, 8101721116, 36040535448, 115738941732, 276500281830, 500602443900, 693602097178, 737001842764, 597227576616, 363697159540, 161963161360, 50290700460, 9984449475, 1056755700, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = -(2+7*x+8*x^2+2*x^3-2*x^4-x^5)*P(n-2,x) - (x+3*x^2+2*x^3-2*x^4-3*x^5-x^6)*d/dx P(n-2,x) + (3+7*x+3*x^2-x^3)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(2 + 7*x + 8*x^2 + 2*x^3 - 2*x^4 - x^5)*P[n - 2, x] - (x + 3*x^2 + 2*x^3 - 2*x^4 - 3*x^5 - x^6)*D[P[n - 2, x], x] + (3 + 7*x + 3*x^2 - x^3)*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] = 1; v[2] = 3 + 4*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 7*x + 8*x^2 + 2*x^3 - 2*x^4 - x^5)*v[i - 2] - (x + 3*x^2 + 2*x^3 - 2*x^4 - 3*x^5 - x^6)*deriv(v[i - 2], x) + (3 + 7*x + 3*x^2 - x^3)*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.