A038331 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 30\,x + 15\,n x\right)\,P_{n}(x) = \left(289 + 1020\,x + 675\,x^{2} + 255\,n x + 225\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 1020\,x + 675\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	289	1020	675															1984
1	4913	26010	34425	13500														78848
2	83521	589560	1170450	918000	253125													3014656
3	1419857	12528150	33162750	39015000	21515625	4556250												112197632
4	24137569	255574260	845650125	1326510000	1097296875	464737500	79734375											4093640704
5	410338673	5068889490	20126472975	39463672500	43526109375	27651881250	9488390625	1366875000										147102629888
6	6975757441	98481281520	456200054100	1073411892000	1479887718750	1253551950000	645210562500	185895000000	23066015625									5222680231936
7	118587876497	1883454509070	9971229753900	27372003246000	45284564193750	47948362087500	32905738687500	14220967500000	3529100390625	384433593750								183618441838592
8	2015993900449	35576362949100	211888632270375	664748650260000	1283062652156250	1630244310975000	1398493894218750	805854825000000	299973533203125	65353710937500	6343154296875							6403555720167424
9	34271896307633	665277987148170	4402574914951125	15538499699827500	34276102279031250	50809281025387500	52303671643781250	37673713068750000	18698350236328125	6110571972656250	1186169853515625	103797070312500						221802281647996928
10	582622237229761	12337882670747880	89812528265002950	352205993196090000	874040608115296875	1480727618454150000	1778324835888562500	1537087493205000000	953615862052734375	415518894140625000	120989325058593750	21174602343750000	1686702392578125					7638104968020361216
11	9904578032905937	227222672519606790	1804415340596877450	7783752449633589000	21462552710386734375	40905100459795893750	56144255533053187500	56616055999717500000	42149821102730859375	22957418901269531250	8912880279316406250	2339793558984375000	372761228759765625	27246730957031250				261713181545754263552
12	168377826559400929	4159922773820493540	35787570921838069425	168412098455708562000	510808754507204278125	1081712656603491412500	1670291602108332328125	1924945903990395000000	1671942903741657421875	1092773139700429687500	530316376619326171875	185623622346093750000	44358586222412109375	6484721967773437500	437893890380859375			8928224131675422982144
13	2862423051509815793	75770021951730418050	701986968082208284875	3578757092183806942500	11841475672667008265625	27583672743389031018750	47324928726402749296875	61357650689693840625000	60906491493446091796875	46442858437268261718750	27046135207585634765625	11833505924563476562500	3770479828905029296875	826802050891113281250	111662942047119140625	7006302246093750000		303412046524374704979968
14	48661191875666868481	1373963064724711580640	13638603951311475249000	74878609928768883720000	268406781913785520687500	682068998745619676100000	1287238061358154780875000	1854364554177413850000000	2070820710777167121093750	1804636784990995312500000	1226091462743882109375000	643742722296253125000000	256392628365541992187500	74963385947460937500000	15186160118408203125000	1905714210937500000000	111662942047119140625	10275869466724347985002496

Technical details

Node id: n-c854448e6f9f291a · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-63bf42bf40951cad

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:            289,            1020,              675;
  1:           4913,           26010,            34425,             13500;
  2:          83521,          589560,          1170450,            918000,            253125;
  3:        1419857,        12528150,         33162750,          39015000,          21515625,           4556250;
  4:       24137569,       255574260,        845650125,        1326510000,        1097296875,         464737500,          79734375;
  5:      410338673,      5068889490,      20126472975,       39463672500,       43526109375,       27651881250,        9488390625,        1366875000;
  6:     6975757441,     98481281520,     456200054100,     1073411892000,     1479887718750,     1253551950000,      645210562500,      185895000000,       23066015625;
  7:   118587876497,   1883454509070,    9971229753900,    27372003246000,    45284564193750,    47948362087500,    32905738687500,    14220967500000,     3529100390625,     384433593750;
  8:  2015993900449,  35576362949100,  211888632270375,   664748650260000,  1283062652156250,  1630244310975000,  1398493894218750,   805854825000000,   299973533203125,   65353710937500,    6343154296875;
  9: 34271896307633, 665277987148170, 4402574914951125, 15538499699827500, 34276102279031250, 50809281025387500, 52303671643781250, 37673713068750000, 18698350236328125, 6110571972656250, 1186169853515625, 103797070312500;

%F Formula

The row polynomials satisfy (17+(15*n+30)*x)*P(n,x) = (289+(255*n+1020)*x+(225*n+675)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1020*x + 675*x^2.

%t Mathematica

P[0, x_] := 289 + 1020*x + 675*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((289 + (255*n + 1020)*x + (225*n + 675)*x^2)*P[n - 1, x])/(17 + (15*n + 30)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1020*x + 675*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (255*n + 1020)*x + (225*n + 675)*x^2)*v[i - 1])/(17 + (15*n + 30)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013610 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.