A038305 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(22 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(403 + 396\,x + 81\,x^{2} + 117\,n + 81\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 403 + 396\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	403	396	81															880
1	6760	10881	5346	729														23716
2	107653	243360	195858	64152	6561													617584
3	1656538	4844385	5475600	2937870	721710	59049												15695152
4	24876631	89453052	130798395	98560800	39661245	7794468	531441											391676032
5	366837484	1567227753	2817771138	2746766295	1552332600	499731687	81841914	4782969										9637291840
6	5333623945	26412298848	56420199108	67626507312	49441793310	22353589440	5996780244	841802544	43046721									234469641472
7	76678687774	432023539545	1069698103344	1523345375916	1369436773068	800957051622	301773457440	69391314252	8523250758	387420489								5652214974208
8	1092263435419	6901081899660	19441059279525	32090943100320	34275270958110	24649861915224	12014355774330	3879944452800	780652285335	85232507580	3486784401							135214152392704
9	15440151087088	108134080106481	341603554033170	641554956224325	794250841732920	678650364970578	406722721601196	169917317379810	48014312603400	8587175138685	843801825042	31381059609						3213750657762304
10	216851407678477	1667536317405504	5839240325749974	12297727945194120	17321983818056775	17155818181431072	12215706569470404	6275150561847024	2293883784627435	576171751240800	92741491497798	8284599736776	282429536481					75961379183472640
11	3028751065922530	25371614698381809	97550874568221984	227730372704248986	359708542396928010	405334421342528535	334538454537905904	204176809804005324	91774076967012726	29820489200156655	6741209489517360	986432227749306	80774847433566	2541865828329				1786845365715841024
12	42099639816323167	381622634306238780	1598411725998053967	4097136731865323328	7173506740183843059	9064655268402585852	8512022848193099235	6021692181682306272	3215784754413083853	1284837077538178164	375738163921973853	77217490516289760	10357538391367713	782894675125332	22876792454961			41855888566696247296
13	582731705083494772	5683451375203627545	25759527815671117650	71928527669912428515	138278364700454662320	193684681984963762593	203954743539058181670	164160440643724056675	101616055565888918340	48236771316196257795	17345300546765405214	4611332011769679105	868696768308259800	107559052525741635	7549341510137130	205891132094649		976825939928167825408
14	8036185203212248849	83913365532023247168	409208499014661183240	1236457335152213647200	2589426996116847426540	3982416903373094274816	4648432367639130302232	4195640438517768308640	2954887931587033020150	1625856889054222693440	694609506953226112248	227065752612201668256	55335984141236149260	9622487279722262400	1106321683121913960	72473678497316448	1853020188851841	22722091290558400626688

Technical details

Node id: n-599179c3856533fe · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e470dede5399547f

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:            403,             396,              81;
  1:           6760,           10881,            5346,             729;
  2:         107653,          243360,          195858,           64152,            6561;
  3:        1656538,         4844385,         5475600,         2937870,          721710,           59049;
  4:       24876631,        89453052,       130798395,        98560800,        39661245,         7794468,          531441;
  5:      366837484,      1567227753,      2817771138,      2746766295,      1552332600,       499731687,        81841914,         4782969;
  6:     5333623945,     26412298848,     56420199108,     67626507312,     49441793310,     22353589440,      5996780244,       841802544,       43046721;
  7:    76678687774,    432023539545,   1069698103344,   1523345375916,   1369436773068,    800957051622,    301773457440,     69391314252,     8523250758,     387420489;
  8:  1092263435419,   6901081899660,  19441059279525,  32090943100320,  34275270958110,  24649861915224,  12014355774330,   3879944452800,   780652285335,   85232507580,   3486784401;
  9: 15440151087088, 108134080106481, 341603554033170, 641554956224325, 794250841732920, 678650364970578, 406722721601196, 169917317379810, 48014312603400, 8587175138685, 843801825042, 31381059609;

%F Formula

The row polynomials satisfy (9*n+22+9*x)*P(n,x) = (117*n+403+(81*n+396)*x+81*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 403 + 396*x + 81*x^2.

%t Mathematica

P[0, x_] := 403 + 396*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((117*n + 403 + (81*n + 396)*x + 81*x^2)*P[n - 1, x])/(9*n + 22 + 9*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 403 + 396*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((117*n + 403 + (81*n + 396)*x + 81*x^2)*v[i - 1])/(9*n + 22 + 9*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.