A038331 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(27 + 10\,x + 10\,n\right)\,P_{n}(x) = \left(629 + 540\,x + 100\,x^{2} + 170\,n + 100\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 629 + 540\,x + 100\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	629	540	100															1269
1	13583	18870	8100	1000														41553
2	280041	543320	377400	108000	10000													1318761
3	5595907	14002050	13583000	6290000	1350000	100000												40920957
4	109328989	335754420	420061500	271660000	94350000	16200000	1000000											1248354909
5	2099968503	7653029230	11751404700	9801435000	4754050000	1320900000	189000000	10000000										37579787433
6	39802851281	167997480240	306121169200	313370792000	196028700000	76064800000	17612000000	2160000000	100000000									1119257792721
7	746406046187	3582256615290	7559886610800	9183635076000	7050842820000	3528516600000	1140972000000	226440000000	24300000000	1000000000								33044255768277
8	13874781550149	74640604618700	179112830764500	251996220360000	229590876900000	141016856400000	58808610000000	16299600000000	2830500000000	270000000000	10000000000							968450880593349
9	256031225357023	1526225970516390	4105233254028500	6567470461365000	6929896059900000	5050999291800000	2585309034000000	924135300000000	224119500000000	34595000000000	2970000000000	100000000000						28207085096966913
10	4695249794145721	30723747042842760	91573558230983400	164209330161140000	197024113840950000	166317505437600000	101019985836000000	44319583440000000	13862029500000000	2988260000000000	415140000000000	32400000000000	1000000000000					817181903283661881
11	85645468872774867	610382473238943730	1997043557784779400	3968187523342614000	5336803230237050000	5122626959864700000	3603545951148000000	1876085451240000000	720193230900000000	200229315000000000	38847380000000000	4906200000000000	351000000000000	10000000000000				23564857741628861997
12	1555018751166232109	11990365642188481380	42726773126726061100	93195366029956372000	138886563316991490000	149430490446637400000	119527962396843000000	72070919022960000000	32831495396700000000	11203005814000000000	2803210410000000000	494421200000000000	57239000000000000	3780000000000000	100000000000000			676776710554169036589
13	28119097035419955143	233252812674934816350	899277423164136103500	2136338656336303055000	3494826226123363950000	4166596899509744700000	3735762261165935000000	2561313479932350000000	1351329731680500000000	547191589945000000000	168045087210000000000	38225596500000000000	6180265000000000000	660450000000000000	40500000000000000	1000000000000000		19367161076277687579993
14	506648880117237395361	4499055525667192822880	18660225013994785308000	47961462568753925520000	85453546253452122200000	111834439235947646400000	111109250653593192000000	85388851683792800000000	51226269598647000000000	24023639674320000000000	8755065439120000000000	2444292177600000000000	509674620000000000000	76064800000000000000	7548000000000000000	432000000000000000	10000000000000000	552456476125005901646241

Technical details

Node id: n-99d7cc44f38135a6 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c94624b66e9d9245

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:             629,              540,              100;
  1:           13583,            18870,             8100,             1000;
  2:          280041,           543320,           377400,           108000,            10000;
  3:         5595907,         14002050,         13583000,          6290000,          1350000,           100000;
  4:       109328989,        335754420,        420061500,        271660000,         94350000,         16200000,          1000000;
  5:      2099968503,       7653029230,      11751404700,       9801435000,       4754050000,       1320900000,        189000000,        10000000;
  6:     39802851281,     167997480240,     306121169200,     313370792000,     196028700000,      76064800000,      17612000000,      2160000000,       100000000;
  7:    746406046187,    3582256615290,    7559886610800,    9183635076000,    7050842820000,    3528516600000,    1140972000000,    226440000000,     24300000000,     1000000000;
  8:  13874781550149,   74640604618700,  179112830764500,  251996220360000,  229590876900000,  141016856400000,   58808610000000,  16299600000000,   2830500000000,   270000000000,   10000000000;
  9: 256031225357023, 1526225970516390, 4105233254028500, 6567470461365000, 6929896059900000, 5050999291800000, 2585309034000000, 924135300000000, 224119500000000, 34595000000000, 2970000000000, 100000000000;

%F Formula

The row polynomials satisfy (10*n+27+10*x)*P(n,x) = (170*n+629+(100*n+540)*x+100*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 629 + 540*x + 100*x^2.

%t Mathematica

P[0, x_] := 629 + 540*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((170*n + 629 + (100*n + 540)*x + 100*x^2)*P[n - 1, x])/(10*n + 27 + 10*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 629 + 540*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((170*n + 629 + (100*n + 540)*x + 100*x^2)*v[i - 1])/(10*n + 27 + 10*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
    
    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
[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.