A038284 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(434 + 912\,x + 288\,x^{2} + 168\,n + 288\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 868 + 1824\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	868	1824	576															3268
1	16856	62496	65664	13824														158840
2	301840	1618176	2999808	2101248	331776													7352848
3	5147744	36220800	97090560	119992320	63037440	7962624												329451488
4	84976192	741275136	2607897600	4660346880	4319723520	1815478272	191102976											14420800576
5	1370375552	14276000256	62267111424	146042265600	195734568960	145142710272	50833391616	4586471424										620252895104
6	21715181824	263112105984	1370496024576	3985095131136	7010028748800	7516207448064	4644566728704	1394287312896	110075314176									26315583996160
7	339431482880	4690479273984	28416107446272	98675713769472	215195137081344	302833241948160	270583468130304	143318059057152	37645757448192	2641807540224								1104339203177984
8	5247905883136	81463555891200	562857512878080	2273288595701760	5920542826168320	10329366579904512	12113329677926400	9277147478753280	4299541771714560	1003886865285120	63403380965376							45930076151071744
9	80412794083328	1385447153147904	10753189377638400	49531461133271040	150037047316316160	312604661221687296	454492129515798528	456845576424652800	306145866798858240	126119891970293760	26502613243527168	1521681143169024						1896019978092443648
10	1222968681238528	23158884695998464	199504390053298176	1032306180253286400	3566265201595514880	8642133925419810816	15005023738640990208	18699104757221425152	16446440751287500800	9796667737563463680	3632252888744460288	693886601285074944	36520347436056576					77774488372878118912
11	18482215434346496	381566228546420736	3612786012575760384	20748456565543010304	80519882059756339200	222534948579560128512	449390964121830162432	668795343779426992128	729265085531635580928	570143279377966694400	305656033411980066816	103023900117115600896	18041051633411948544	876488338465357824				3173008267973248409600
12	277800170638950400	6210024385940422656	64103126395798683648	404632033408485163008	1742870351505612865536	5410936074415625994240	12461957120455367196672	21570766277847847796736	28089404438735933669376	27225896526514395021312	19156814187099680931840	9336402475129572950016	2884669203279236825088	466291796063570362368	21035720123168587776			128842267155530875420672
13	4155890552758697984	100008061430022144000	1117804389469276078080	7692375167495842037760	36416883006763664670720	125486665308404126318592	324656164464937559654400	640900651909133170114560	970684482503153150853120	1123576177549437346775040	980132274954518220767232	626950282486898648678400	280092074253887188500480	79883147167732712079360	11990360470206095032320	504857282956046106624		5210188364866975828508672
14	61916102032009265152	1595861972259340025856	19201547794564251648000	143078961852067337994240	738468016079600835624960	2796816614919449446711296	8031146579737864084389888	17809709593505146129612800	30763231291638392165498880	41415871253467867769733120	43145325217898394116161536	34215526689321363343147008	20062409039580756757708800	8273488962576360029552640	2191080608029240102748160	306953228037276032827392	12116574790945106558976	209926081957303578859208704

Technical details

Node id: n-48f947c3236c9f1b · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-29ed0a4e339957b1

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            868,             1824,               576;
  1:          16856,            62496,             65664,             13824;
  2:         301840,          1618176,           2999808,           2101248,             331776;
  3:        5147744,         36220800,          97090560,         119992320,           63037440,            7962624;
  4:       84976192,        741275136,        2607897600,        4660346880,         4319723520,         1815478272,          191102976;
  5:     1370375552,      14276000256,       62267111424,      146042265600,       195734568960,       145142710272,        50833391616,         4586471424;
  6:    21715181824,     263112105984,     1370496024576,     3985095131136,      7010028748800,      7516207448064,      4644566728704,      1394287312896,       110075314176;
  7:   339431482880,    4690479273984,    28416107446272,    98675713769472,    215195137081344,    302833241948160,    270583468130304,    143318059057152,     37645757448192,      2641807540224;
  8:  5247905883136,   81463555891200,   562857512878080,  2273288595701760,   5920542826168320,  10329366579904512,  12113329677926400,   9277147478753280,   4299541771714560,   1003886865285120,    63403380965376;
  9: 80412794083328, 1385447153147904, 10753189377638400, 49531461133271040, 150037047316316160, 312604661221687296, 454492129515798528, 456845576424652800, 306145866798858240, 126119891970293760, 26502613243527168, 1521681143169024;

%F Formula

The row polynomials satisfy (12*n+19+12*x)*P(n,x) = (168*n+434+(288*n+912)*x+288*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 868 + 1824*x + 576*x^2.

%t Mathematica

P[0, x_] := 868 + 1824*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((168*n + 434 + (288*n + 912)*x + 288*x^2)*P[n - 1, x])/(12*n + 19 + 12*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 868 + 1824*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((168*n + 434 + (288*n + 912)*x + 288*x^2)*v[i - 1])/(12*n + 19 + 12*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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

Related ?

Same fingerprint ?

A038271 → x ↦ x/(x+1) → D+1

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.