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	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

All 15 rows

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.