A038331 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 44\,x + 27\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 918\,x + 729\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	289	918	729															1936
1	4913	23409	37179	19683														85184
2	83521	530604	1264086	1338444	531441													3748096
3	1419857	11275335	35815770	56883870	45172485	14348907												164916224
4	24137569	230016834	913302135	1934051580	2303796735	1463588514	387420489											7256313856
5	410338673	4562000541	21736590813	57538034505	91383937155	87083516583	46103038191	10460353203										319277809664
6	6975757441	88633153368	492696058428	1565034538536	3107053863270	3947786085096	3135006596988	1422608035608	282429536481									14048223625216
7	118587876497	1695109058163	10768928134212	39908380732668	95075848216062	151002817754922	159885336446388	108829514724012	43211719081593	7625597484987								618121839509504
8	2015993900449	32018726654190	228839722852005	969203532079080	2693815699455090	5134095803667348	6795126798971490	6167005834360680	3672996121935405	1296351572447790	205891132094649							27197360938418176
9	34271896307633	598750188433353	4754780908147215	22655132562348495	71963362256871690	160012652547632346	254137742281533726	288307522756361790	228950091600640245	121208872023868365	38501641701699363	5559060566555523						1196683881290399744
10	582622237229761	11104094403673092	96997530526203186	513516338079899220	1835065737550228095	4663225874245285512	8640683237572146684	11762946928459561032	11676454671632652495	8242203297623048820	3927167453573335026	1134048355577326692	150094635296999121					52654090776777588736
11	9904578032905937	204500405267646111	1948768567844627646	11348711071565772762	45061058666511156555	128821614776026012269	272798713643349202452	433268545198260498012	516099296486163240279	455381732193673447305	289301335746569013582	125312343291294599466	33170914400636805741	4052555153018976267				2316779994178213904384
12	168377826559400929	3743930496438444186	38650576595585114979	245544839548423083396	1072453196262965526009	3406616035188243435558	8115761730889638772947	14731130536740856932408	20471938760617808531067	21676170452418856091718	17213429476920856308129	9941445901109371557636	3947338813675779883179	964508126418516351546	109418989131512359209			101938319743841411792896
13	2862423051509815793	68193019756557376245	758145925528784947665	5217827840403990522165	24861415004277837193845	86868708897300207606729	229946582375206431900165	469554785858614814720505	745763483422505882203155	921237244227801383898015	877884903322963671714579	633767176195722436799295	335523799162441290070215	122974786118360834822115	27901842228535651598295	2954312706550833698643		4485286068729022118887424
14	48661191875666868481	1236566758252240422576	14729692267416393268920	109173013276145032463760	563525406763630976393820	2148026256369605133548208	6254547040605614947684488	14190989083727025511553040	25355958436365199994907270	35796647204280282345751440	39797448950641019784394248	34476934385047300561881648	22815618343046007724774620	11149713941398049023871760	3794650543080848617368120	803573056181826766030896	79766443076872509863361	197352587024076973231046656

Technical details

Node id: n-e09f58916fb51b22 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-df32274ca8b60243

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,             918,              729;
  1:           4913,           23409,            37179,             19683;
  2:          83521,          530604,          1264086,           1338444,            531441;
  3:        1419857,        11275335,         35815770,          56883870,          45172485,           14348907;
  4:       24137569,       230016834,        913302135,        1934051580,        2303796735,         1463588514,          387420489;
  5:      410338673,      4562000541,      21736590813,       57538034505,       91383937155,        87083516583,        46103038191,        10460353203;
  6:     6975757441,     88633153368,     492696058428,     1565034538536,     3107053863270,      3947786085096,      3135006596988,      1422608035608,       282429536481;
  7:   118587876497,   1695109058163,   10768928134212,    39908380732668,    95075848216062,    151002817754922,    159885336446388,    108829514724012,     43211719081593,      7625597484987;
  8:  2015993900449,  32018726654190,  228839722852005,   969203532079080,  2693815699455090,   5134095803667348,   6795126798971490,   6167005834360680,   3672996121935405,   1296351572447790,   205891132094649;
  9: 34271896307633, 598750188433353, 4754780908147215, 22655132562348495, 71963362256871690, 160012652547632346, 254137742281533726, 288307522756361790, 228950091600640245, 121208872023868365, 38501641701699363, 5559060566555523;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (17+44*x+27*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 918*x + 729*x^2.

%t Mathematica

P[0, x_] := 289 + 918*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 44*x + 27*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 289 + 918*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 44*x + 27*x^2)*v[i - 1])/(1 + 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) 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.