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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	25733569199849	274844843175400	1307339012178000	3638014399680000	6535885742400000	7880106124800000	6404979840000000	3418060800000000	1116288000000000	189440000000000	10240000000000							30800932331433249
9	477790554406413	5661385223966780	30232932749294000	95871527559720000	200090791982400000	287578972665600000	288937224576000000	201299366400000000	93996672000000000	27287040000000000	4167680000000000	204800000000000						1235806183711387193
10	8807877351061681	114669733057539120	679366226876013600	2418634619943520000	5752291653583200000	9604358015155200000	11503158906624000000	9906419128320000000	6038980992000000000	2506577920000000000	654888960000000000	90931200000000000	4096000000000000					49283181232910534401
11	161386359712643797	2290048111276037060	14907065297480085600	58878406329254512000	157211250296328800000	299119165986326400000	416188847323392000000	427260187960320000000	321958621670400000000	174459450880000000000	65171025920000000000	15479193600000000000	1970176000000000000	81920000000000000				1955136745734490478457
12	2941659675773063289	45188180719540263160	320606735578645188400	1391326094431474656000	4121488443047815840000	8803830016594412800000	13958894412695232000000	16647553892935680000000	14954106578611200000000	10016490451968000000000	4884864624640000000000	1658898841600000000000	361181184000000000000	42434560000000000000	1638400000000000000			77211444076497773810849
13	53375771019330094493	882497902731918986700	6778227107931039474000	32060673557864518840000	104349457082360599200000	247289306582868950400000	440191500829720640000000	598238331972652800000000	624283270985088000000000	498470219287040000000000	300494713559040000000000	133223580672000000000000	41472471040000000000000	8334950400000000000000	909312000000000000000	32768000000000000000		3037064656750317796995193
14	964636568358807922241	17080246726185630237760	141199664437107037872000	723010891512644210560000	2564853884629161507200000	6678365253271078348800000	13188763017753010688000000	20123040037930086400000000	23929533278906112000000000	22196738523914240000000000	15951047017185280000000000	8741664394444800000000000	3552628817920000000000000	1020860825600000000000000	190513152000000000000000	19398656000000000000000	655360000000000000000	119040317658798064630592001

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.