A200536 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 - 5\,x + 6\,x^{2} + 7\,n + 12\,n x\right)\,P_{n}(x) = \left(12 + 84\,x + 219\,x^{2} + 252\,x^{3} + 108\,x^{4} + 42\,n + 219\,n x + 378\,n x^{2} + 216\,n x^{3}\right)\,P_{n-1}(x)\)

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

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	27	57	18																											102
2	288	1566	2925	2052	324																									7155
3	2484	22032	78489	141453	131706	55404	5832																							437400
4	19440	239760	1267488	3729564	6619401	7140312	4426488	1329696	104976																					24877125
5	143856	2274480	15969960	65396160	172288215	303367761	358237890	275649480	128070720	29918160	1889568																			1353206250
6	1026432	19805472	173525328	911483280	3190570560	7819412922	13704801237	17207849628	15220863900	9101419200	3381381936	646232256	34012224																	71412384375
7	7138368	162642816	1708484400	10957079952	47875468284	150536948688	350581136661	612644517765	804148978122	784404317844	554577342984	271159621488	83925162720	13570877376	612220032															3686871937500
8	48708864	1279867392	15677535744	118766487168	622342447776	2389698648288	6947669018304	15577633391544	27168173753553	36879126013296	38684387787696	30847397894208	18162714683232	7511123499264	1992163984128	279172334592	11019960576													187208396015625
9	327525120	9750170880	136457042688	1192483689984	7288247064096	33063299956512	115296183526608	315714478512384	687409598738835	1197202755113001	1668482106359922	1851259273447248	1616916470097600	1090276788692448	549970417476288	197047915059456	45721816429824	5653239775488	198359290368											9382839967968750
10	2176782336	72408245760	1139863000320	11287976901120	78838747994880	412639464704832	1678523576703840	5430681609706080	14183098343336160	30168123834331350	52487643067933437	74715808518579060	86695090618256820	81330768094060800	60848464034950560	35540844022336896	15680880451520640	4956006869844480	1022046243621120	113064795509760	3570467226624									465358595185546875
11	14330483712	526781325312	9213185887488	101966606818560	801182765243520	4752568738139904	22094613097295904	82481963224817376	251299022956437060	631584800000597520	1318079792510388369	2291484986671467069	3319235129077505946	3994675707187283580	3970094070806705160	3225838292814357792	2110667906783126592	1087326236491406208	426298909912128000	120525584318726400	22367191941186048	2238682951093248	64268410079232							22882022631562500000
12	93601640448	3768010223616	72486851788800	886686707377152	7741364351871744	51327516095387136	268473746269251072	1135993777767319680	3955824316854576144	11472117171584948208	27930629032522688736	57375148217464750932	99687849532938816345	146518956462101081544	181774252803124110888	189465327519564732768	164663094428370598896	117993205147433806080	68595507214946249472	31606904157719261184	11149789894274541312	2851610778018883584	481049049443051520	43959592494194688	1156831381426176					1117030881589892578125
13	607322271744	26571981975552	557629596186624	7470886766174208	71752623242916096	525869779614296832	3056627450643376896	14457251067247093248	56645779788193589136	186207387170073386832	518177425949215316856	1228277417879497572288	2489688967493939723343	4323759481189899523521	6434110140970378953762	8189906263330826667672	8884719874466519630976	8166446356404480631344	6305626086774704303328	4041073410119910717696	2113594626537862490880	880880944204044875520	282432782035709001216	65942666430206072832	10196311795890315264	857212053636796416	20822964865671168			54196683514176269531250
14	3918208204800	185135337676800	4208880480473088	61293921911599104	642246417344802816	5155810997039119872	32969668396598938368	172383581722752271872	750613327584716179584	2758264310153907146592	8636837906541623963184	23206376841249225392496	53765975626304501484288	107746023046694937344370	187049319599958711738597	281322462241509877560204	366054654268401808735116	410877164109863736492480	395990150981745327383376	325503727742642307034176	226103471656969100306112	131054045414100212371968	62294500073104014014208	23699009539887426081792	6964009522255282258944	1495699684324582883328	213466624320427978752	16616725962805592064	374813367582081024	2615713981056331787109375

Technical details

Node id: n-41aa178d33268925 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-19ef812d44a90303

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:        27,         57,           18;
  2:       288,       1566,         2925,          2052,           324;
  3:      2484,      22032,        78489,        141453,        131706,          55404,            5832;
  4:     19440,     239760,      1267488,       3729564,       6619401,        7140312,         4426488,         1329696,          104976;
  5:    143856,    2274480,     15969960,      65396160,     172288215,      303367761,       358237890,       275649480,       128070720,         29918160,          1889568;
  6:   1026432,   19805472,    173525328,     911483280,    3190570560,     7819412922,     13704801237,     17207849628,     15220863900,       9101419200,       3381381936,        646232256,         34012224;
  7:   7138368,  162642816,   1708484400,   10957079952,   47875468284,   150536948688,    350581136661,    612644517765,    804148978122,     784404317844,     554577342984,     271159621488,      83925162720,      13570877376,       612220032;
  8:  48708864, 1279867392,  15677535744,  118766487168,  622342447776,  2389698648288,   6947669018304,  15577633391544,  27168173753553,   36879126013296,   38684387787696,   30847397894208,   18162714683232,    7511123499264,   1992163984128,    279172334592,    11019960576;
  9: 327525120, 9750170880, 136457042688, 1192483689984, 7288247064096, 33063299956512, 115296183526608, 315714478512384, 687409598738835, 1197202755113001, 1668482106359922, 1851259273447248, 1616916470097600, 1090276788692448, 549970417476288, 197047915059456, 45721816429824, 5653239775488, 198359290368;

%F Formula

The row polynomials satisfy (7*n-5+(12*n-5)*x+6*x^2)*P(n,x) = (42*n+12+(219*n+84)*x+(378*n+219)*x^2+(216*n+252)*x^3+108*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((42*n + 12 + (219*n + 84)*x + (378*n + 219)*x^2 + (216*n + 252)*x^3 + 108*x^4)*P[n - 1, x])/(7*n - 5 + (12*n - 5)*x + 6*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; for(n = 1, N, my(i = n + 1); v[i] = ((42*n + 12 + (219*n + 84)*x + (378*n + 219)*x^2 + (216*n + 252)*x^3 + 108*x^4)*v[i - 1])/(7*n - 5 + (12*n - 5)*x + 6*x^2)); 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 basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    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.