A038331 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 20\,x + 10\,n x\right)\,P_{n}(x) = \left(289 + 680\,x + 300\,x^{2} + 170\,n x + 100\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 680\,x + 300\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	289	680	300								1269
1	4913	17340	15300	4000							41553
2	83521	393040	520200	272000	50000						1318761
3	1419857	8352100	14739000	11560000	4250000	600000					40920957
4	24137569	170382840	375844500	393040000	216750000	61200000	7000000				1248354909
5	410338673	3379259660	8945099100	11692940000	8597750000	3641400000	833000000	80000000			37579787433
6	6975757441	65654187680	202755579600	318047968000	292323500000	165076800000	56644000000	10880000000	900000000		1119257792721
7	118587876497	1255636339380	4431657668400	8110223184000	8945099100000	6314187600000	2888844000000	832320000000	137700000000	10000000000	33044255768277

All 15 rows

Technical details

Node id: n-a03b2910f3829be2 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-d6da840874f40ef5

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,             680,              300;
  1:           4913,           17340,            15300,             4000;
  2:          83521,          393040,           520200,           272000,            50000;
  3:        1419857,         8352100,         14739000,         11560000,          4250000,           600000;
  4:       24137569,       170382840,        375844500,        393040000,        216750000,         61200000,          7000000;
  5:      410338673,      3379259660,       8945099100,      11692940000,       8597750000,       3641400000,        833000000,         80000000;
  6:     6975757441,     65654187680,     202755579600,     318047968000,     292323500000,     165076800000,      56644000000,      10880000000,       900000000;
  7:   118587876497,   1255636339380,    4431657668400,    8110223184000,    8945099100000,    6314187600000,    2888844000000,     832320000000,    137700000000,     10000000000;
  8:  2015993900449,  23717575299400,   94172725453500,  196962563040000,  253444474500000,  214682378400000,  122775870000000,   47164800000000,  11704500000000,   1700000000000,   110000000000;
  9: 34271896307633, 443518658098780, 1956699962200500, 4603999911060000, 6770588104500000, 6690934126800000, 4591817538000000, 2204954400000000, 729580500000000, 158950000000000, 20570000000000, 1200000000000;

%F Formula

The row polynomials satisfy (17+(10*n+20)*x)*P(n,x) = (289+(170*n+680)*x+(100*n+300)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 680*x + 300*x^2.

%t Mathematica

P[0, x_] := 289 + 680*x + 300*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((289 + (170*n + 680)*x + (100*n + 300)*x^2)*P[n - 1, x])/(17 + (10*n + 20)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 289 + 680*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (170*n + 680)*x + (100*n + 300)*x^2)*v[i - 1])/(17 + (10*n + 20)*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_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.