A272866 → D → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 15\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 6\,x\)

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	5	6																											11
2	50	210	270	108																									638
3	375	2700	7425	9720	6075	1458																							27753
4	2500	25500	108000	245700	324000	247860	102060	17496																					1073116
5	15625	206250	1181250	3847500	7846875	10388250	8930250	4811400	1476225	196830																			38900455
6	93750	1518750	10968750	46575000	129093750	245126250	325316250	301806000	191909250	79716150	19486170	2125764																	1353735834
7	546875	10500000	91546875	479587500	1683281250	4178081250	7542871875	10027395000	9816896250	6992385750	3523749075	1190427840	241805655	22320522															45801395717
8	3125000	69375000	708750000	4417875000	18781875000	57663900000	132040125000	229580325000	305578575000	311385060000	240978969000	139131253800	58033357200	16517186280	2869781400	229582512													1517989115192
9	17578125	442968750	5189062500	37513125000	187375781250	686034562500	1905970500000	4103413425000	6929953734375	9232680206250	9704112660000	8001907137000	5114189603250	2480926020300	882457780500	216955473840	32930741565	2324522934											49524394883139
10	97656250	2753906250	36386718750	299320312500	1718085937500	7309364062500	23896392187500	61423261875000	125949671718750	207826030968750	277089277781250	298517052712500	258818410012500	179038487092500	97393206262500	40722198066000	12623450933250	2731314447450	368049464550	23245229340									1595786057345590
11	537109375	16757812500	246533203125	2273906250000	14748486328125	71513191406250	269043714843750	804481115625000	1942781298046875	3829303046250000	6200022288796875	8271294579900000	9092216474859375	8210534266068750	6052099930593750	3603441014910000	1705537990220625	626458930713000	172063124677125	33240677956200	4027235983155	230127770466							50905575229324321
12	2929687500	100195312500	1624218750000	16601835937500	120076171875000	653736445312500	2782530351562500	9492741843750000	26402745445312500	60576616082812500	115554205985625000	184167855979312500	245815310914875000	274775530551637500	256634685728437500	199302013557900000	127688950418287500	66705091244797500	27934954359345000	9147578876023500	2255252150566800	393518487496860	43305862260420	2259436291848					1610467289073169428
13	15869140625	590332031250	10454589843750	117297070312500	936028564453125	5654178105468750	26865999316406250	103009066875000000	324401597050781250	849722183578125000	1867715395192968750	3465953740628437500	5450880768812578125	7278502855319718750	8251906200579843750	7928927639404200000	6432494387600531250	4379459320729350000	2480239335419381250	1153867033554592500	433229910887899125	127968299961880950	28617957310427550	4552764128073720	458947996781625	22029503845518			50595513998382072863
14	85449218750	3435058593750	65983886718750	806203125000000	7036281738281250	46702915136718750	245072073339843750	1043455459921875000	3671163980859375000	10814156204414062500	26927410572562500000	57069383843489062500	103441852241782031250	160837717895554218750	214840770040624218750	246539738203872750000	242699199436921500000	204340835108622712500	146445961820783625000	88725602092214587500	45011409774727646250	18871023731666856750	6420274521200016750	1727113101488611200	353389957521851250	51659186517739710	4804126415541810	213516729579636	1580136821795624737106

Technical details

Node id: n-57c3bb526ceab4ae · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9de613d7fd734864

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:        5,         6;
  2:       50,       210,        270,         108;
  3:      375,      2700,       7425,        9720,         6075,         1458;
  4:     2500,     25500,     108000,      245700,       324000,       247860,        102060,         17496;
  5:    15625,    206250,    1181250,     3847500,      7846875,     10388250,       8930250,       4811400,       1476225,        196830;
  6:    93750,   1518750,   10968750,    46575000,    129093750,    245126250,     325316250,     301806000,     191909250,      79716150,      19486170,       2125764;
  7:   546875,  10500000,   91546875,   479587500,   1683281250,   4178081250,    7542871875,   10027395000,    9816896250,    6992385750,    3523749075,    1190427840,     241805655,      22320522;
  8:  3125000,  69375000,  708750000,  4417875000,  18781875000,  57663900000,  132040125000,  229580325000,  305578575000,  311385060000,  240978969000,  139131253800,   58033357200,   16517186280,   2869781400,    229582512;
  9: 17578125, 442968750, 5189062500, 37513125000, 187375781250, 686034562500, 1905970500000, 4103413425000, 6929953734375, 9232680206250, 9704112660000, 8001907137000, 5114189603250, 2480926020300, 882457780500, 216955473840, 32930741565, 2324522934;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n+15*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 6*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 5 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 15*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] = 5 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 15*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) 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
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272866 → basis A013610 → D

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.