A038331 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(37 + 20\,x + 20\,n\right)\,P_{n}(x) = \left(969 + 1480\,x + 400\,x^{2} + 340\,n + 400\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 969 + 1480\,x + 400\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	969	1480	400								2849
1	22253	58140	44400	8000							132793
2	476561	1780240	2325600	1184000	160000						5926401
3	9771957	47656100	89012000	77520000	29600000	3200000					256760057
4	194520409	1172634840	2859366000	3560480000	2325600000	710400000	64000000				10887001249
5	3789598333	27232857260	82084438800	133437080000	124616800000	65116800000	16576000000	1280000000			454133574393
6	72629945121	606335733280	2178628580800	4377836736000	5337483200000	3987737600000	1736448000000	378880000000	25600000000		18701579795201
7	1374224215877	13073390121780	54570215995200	130717714848000	197002653120000	192149395200000	119632128000000	44651520000000	8524800000000	512000000000	762208041500857

All 15 rows

Technical details

Node id: n-35f903b481938239 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-73011d80f84eea5c

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:             969,             1480,               400;
  1:           22253,            58140,             44400,              8000;
  2:          476561,          1780240,           2325600,           1184000,             160000;
  3:         9771957,         47656100,          89012000,          77520000,           29600000,            3200000;
  4:       194520409,       1172634840,        2859366000,        3560480000,         2325600000,          710400000,           64000000;
  5:      3789598333,      27232857260,       82084438800,      133437080000,       124616800000,        65116800000,        16576000000,         1280000000;
  6:     72629945121,     606335733280,     2178628580800,     4377836736000,      5337483200000,      3987737600000,      1736448000000,       378880000000,       25600000000;
  7:   1374224215877,   13073390121780,    54570215995200,   130717714848000,    197002653120000,    192149395200000,    119632128000000,     44651520000000,     8524800000000,      512000000000;
  8:  25733569199849,  274844843175400,  1307339012178000,  3638014399680000,   6535885742400000,   7880106124800000,   6404979840000000,   3418060800000000,  1116288000000000,   189440000000000,   10240000000000;
  9: 477790554406413, 5661385223966780, 30232932749294000, 95871527559720000, 200090791982400000, 287578972665600000, 288937224576000000, 201299366400000000, 93996672000000000, 27287040000000000, 4167680000000000, 204800000000000;

%F Formula

The row polynomials satisfy (20*n+37+20*x)*P(n,x) = (340*n+969+(400*n+1480)*x+400*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 969 + 1480*x + 400*x^2.

%t Mathematica

P[0, x_] := 969 + 1480*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((340*n + 969 + (400*n + 1480)*x + 400*x^2)*P[n - 1, x])/(20*n + 37 + 20*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 969 + 1480*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((340*n + 969 + (400*n + 1480)*x + 400*x^2)*v[i - 1])/(20*n + 37 + 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 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_A013611 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.