A272867 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 18\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 18 + 18\,x\)

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	18	18																											36
2	216	864	972	324																									2376
3	1944	13608	34992	40824	21870	4374																							117612
4	15552	155520	629856	1329696	1574640	1049760	367416	52488																					5174928
5	116640	1516320	8398080	25894080	48813840	58261680	44089920	20470320	5314410	590490																			213465780
6	839808	13436928	94478400	384212160	1001471040	1753519104	2103089184	1728954720	956593800	340122240	70150212	6377292																	8453244888
7	5878656	111694464	952342272	4814619264	16070775840	37339753248	62060971392	74679506496	65086642152	40623350040	17677853424	5089079016	870500358	66961566															325449928188
8	40310784	886837248	8888527872	53754430464	219038722560	636164637696	1358992422144	2172836918016	2620913956992	2385617391360	1626362515008	816395412672	292488120288	70711413696	10331213040	688747536													12274111577376
9	272097792	6802444800	78364164096	551814322176	2657034938880	9273909044736	24273299828736	48627412701696	75450915073728	91176398815680	85823452965888	62598886051968	34973222383008	14666190031584	4463084033280	929809173600	118550669634	6973568802											455676392310084
10	1813985280	50791587840	661197634560	5314069877760	29533494343680	120484902297600	373870529141760	902387838366720	1718287352812800	2603135087262720	3150193480156800	3045558954487680	2343367066584960	1423227908390400	672809918016960	242122308805440	64017361602360	11715595587360	1324978072380	69735688020									16708134384703080
11	11972302848	371141388288	5387536281600	48667411077120	306685502830080	1432572834958848	5145528151830528	14551196742973440	32909293966728960	60136420761457920	89344396489683456	108245557370624256	106955205391869120	85943005763186880	55819792004232960	29009599907916672	11877354489291192	3741877547777160	874485527770800	142679217688920	14498049539358	690383311398							606505278164721804
12	78364164096	2664381579264	42669287350272	427985882210304	3015942810439680	15876853920423936	64821435780777984	210348919315003392	551658609316767744	1182705367261777920	2088769743648986112	3053345746790366208	3702819091014111744	3725521906874600448	3103079677406818560	2129782808064409344	1195434603617829696	542541867561986688	195814799378437536	54844050257457120	11482455235171536	1690058346302304	155901104137512	6778308875544					21834190013929984944
13	509367066624	18846581465088	330069859172352	3639937058095104	28365378522624000	166183297637879808	760343426425110528	2786085298996826112	8319593886064141824	20495829732199810560	42018429436161564672	72109134686094944256	103974071154581481984	126190985700666152448	128920181224586618880	110677480553878977024	79556116535488356192	47602879288328583648	23507849998692423936	9464658123282373440	3053289232988794800	769622746347016848	145923433472711232	19562199414819984	1652212788413850	66088511536554			780572292997996961748
14	3291294892032	131651795681280	2503029765390336	30100537435078656	257041902830469120	1658931112200241152	8409313578331201536	34352061852094930944	115134050269704001536	320710315728195993600	749443538061231197184	1479098929061681980416	2476895848218240583680	3529789207712188710912	4286935121916185625600	4437296787185045941248	3909029363296649085312	2922472752073185991680	1845742743386192274624	978228636335047056960	431043018651804791232	155894163149223442944	45455149526749548768	10414492001976325536	1804216364947924200	222057398762821440	17294855095950516	640550188738908	27740338412698045871352

Technical details

Node id: n-22d01599550a0f21 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-b643eea373acac66

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:        18,         18;
  2:       216,        864,         972,          324;
  3:      1944,      13608,       34992,        40824,         21870,          4374;
  4:     15552,     155520,      629856,      1329696,       1574640,       1049760,         367416,          52488;
  5:    116640,    1516320,     8398080,     25894080,      48813840,      58261680,       44089920,       20470320,        5314410,         590490;
  6:    839808,   13436928,    94478400,    384212160,    1001471040,    1753519104,     2103089184,     1728954720,      956593800,      340122240,       70150212,        6377292;
  7:   5878656,  111694464,   952342272,   4814619264,   16070775840,   37339753248,    62060971392,    74679506496,    65086642152,    40623350040,    17677853424,     5089079016,      870500358,       66961566;
  8:  40310784,  886837248,  8888527872,  53754430464,  219038722560,  636164637696,  1358992422144,  2172836918016,  2620913956992,  2385617391360,  1626362515008,   816395412672,   292488120288,    70711413696,   10331213040,    688747536;
  9: 272097792, 6802444800, 78364164096, 551814322176, 2657034938880, 9273909044736, 24273299828736, 48627412701696, 75450915073728, 91176398815680, 85823452965888, 62598886051968, 34973222383008, 14666190031584, 4463084033280, 929809173600, 118550669634, 6973568802;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+18*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 18 + 18*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 18 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 18*n*x + 9*n*x^2)*P[n - 1, x])/(n - 1)]];
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] = 18 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 18*n*x + 9*n*x^2)*v[i - 1])/(n - 1)); 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, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

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.