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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	2015993900449	23717575299400	94172725453500	196962563040000	253444474500000	214682378400000	122775870000000	47164800000000	11704500000000	1700000000000	110000000000							968450880593349
9	34271896307633	443518658098780	1956699962200500	4603999911060000	6770588104500000	6690934126800000	4591817538000000	2204954400000000	729580500000000	158950000000000	20570000000000	1200000000000						28207085096966913
10	582622237229761	8225255113831920	39916679228890200	104357331317360000	172649996664750000	194992937409600000	156121796292000000	89962139520000000	37208605500000000	10808600000000000	2098140000000000	244800000000000	13000000000000					817181903283661881
11	9904578032905937	151481781679737860	801962373598612200	2306297022113656000	4239516584767750000	5386679895940200000	4928988140076000000	3313605472320000000	1644620363100000000	597175150000000000	154562980000000000	27050400000000000	2873000000000000	140000000000000				23564857741628861997
12	168377826559400929	2773281849213662360	15905587076372475300	49899881023913648000	100900494717472450000	142447757248196400000	146637397167261000000	112662586058880000000	65236607736300000000	28425537140000000000	9196497310000000000	2145998400000000000	341887000000000000	33320000000000000	1500000000000000			676776710554169036589
13	2862423051509815793	50513347967820278700	311994208036537015500	1060372471758165020000	2339056922995952250000	3632417809829008200000	4154726253072395000000	3591119930626800000000	2376476424679500000000	1208085328450000000000	469021362810000000000	136807398000000000000	29060395000000000000	4248300000000000000	382500000000000000	16000000000000000		19367161076277687579993
14	48661191875666868481	915975376483141053760	6061601756138433444000	22186254793709298880000	53018623587908251000000	89819785843044566400000	113008554083569144000000	108531624570054400000000	80800198439103000000000	46942744191200000000000	21262301780720000000000	7442322451200000000000	1976106860000000000000	385179200000000000000	52020000000000000000	4352000000000000000	170000000000000000	552456476125005901646241

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.