A038321 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(39 + 42\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1521 + 3276\,x + 1764\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1521	3276	1764															6561
1	59319	191646	206388	74088														531441
2	2313441	9965592	16098264	11557728	3111696													43046721
3	90224199	485822610	1046387160	1126878480	606780720	130691232												3486784401
4	3518743761	22736498148	61213648860	87896521440	70993344240	30581748288	5489031744											282429536481
5	137231006679	1034510665734	3342265227756	5998937588280	6460394325840	4174408641312	1498505666112	230539333248										22876792454961
6	5352009260481	46109618244144	173797791843312	374333705508672	503910757415520	434138498696448	233766883913472	71928271973376	9682651996416									1853020188851841
7	208728361158759	2023059500461818	8714717848143216	21898521772257312	35374535170569504	38095653260613312	27350725417876224	12623411731327488	3398610850742016	406671383849472								150094635296999121
8	8140406085191601	87665911686678780	424842495096981780	1220060498740050240	2299344786087017760	2971460954327838336	2666695728242931840	1641043525072573440	662729115894693120	158601839701294080	17080198121677824							12157665459056928801
9	317475837322472439	3760867611358519662	20250825599622798180	65425744244935194120	140916987604475802720	212459458234440441024	228802493483243551872	176001918064033501440	94770263572941116160	34020094615927580160	7327404994199786496	717368321110468608						984770902183611232881
10	12381557655576425121	160007822010526109256	947738638062346954824	3402138700736630094240	8243643774861834459120	14204432350531160914176	17846594491692997046016	16473779530793535734784	11088120838034110590720	5307134760084702504960	1714612768642750040064	335728374279699308544	30129469486639681536					79766443076872509863361
11	482880748567480579719	6760330479944728116066	43682135408873627826888	172488432127347145777968	464391932650550007863760	900205900214912322935904	1292603343898335643190016	1392034370352053769589248	1124335452976658813899008	672679330840736042503680	289769557900624756770816	85107142879903774715904	15275641029726318538752	1265437718438866624512				6461081889226673298932241
12	18832349194131742609041	283933880157678580874772	1987537161103750066123404	8561698540139231054070048	25355799522720030429361296	54612491279704680924778176	88220178221061407647718592	108578680887460194027961344	102314526220875952064809728	73456582927808375841401856	39553544653435279299216384	15489500004142486998294528	4170250001115284961079296	690928994267621176983552	53148384174432398229504			523347633027360537213511521
13	734461618571137961752599	11864379992302997843695830	89439172249668752975553180	417382803831787513885914840	1348467520071928891016032560	3194830739862723834099523296	5734311584368991497101708480	7939816039895526688294673280	8550571119887490279701955840	7162016835461316644536680960	4627764724451927678008316928	2265339375605838723500574720	813198750217480567410462720	202096730823279194267688960	31091804742042952964259840	2232232135326160725639168		42391158275216203514294433201
14	28644003124274380508351361	493558207679804710297746528	3986431677413807275481798880	20034374583925800666523912320	70120311043740302332833693120	181234034697667242952554776064	357821042864625069419146609152	550493912099423183721764014080	666944547351224241816752555520	638442643618265940884412702720	481287531343000478512864960512	282714354075608672692871725056	126859005033926968516032184320	42036120011242072407679303680	9700643079517401324849070080	1392912852443524292798840832	93753749683698750476845056	3433683820292512484657849089281

Technical details

Node id: n-23763ec4ded1b991 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-b21306f08ea8390f

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:               1521,                3276,                 1764;
  1:              59319,              191646,               206388,                74088;
  2:            2313441,             9965592,             16098264,             11557728,               3111696;
  3:           90224199,           485822610,           1046387160,           1126878480,             606780720,             130691232;
  4:         3518743761,         22736498148,          61213648860,          87896521440,           70993344240,           30581748288,            5489031744;
  5:       137231006679,       1034510665734,        3342265227756,        5998937588280,         6460394325840,         4174408641312,         1498505666112,          230539333248;
  6:      5352009260481,      46109618244144,      173797791843312,      374333705508672,       503910757415520,       434138498696448,       233766883913472,        71928271973376,        9682651996416;
  7:    208728361158759,    2023059500461818,     8714717848143216,    21898521772257312,     35374535170569504,     38095653260613312,     27350725417876224,     12623411731327488,     3398610850742016,      406671383849472;
  8:   8140406085191601,   87665911686678780,   424842495096981780,  1220060498740050240,   2299344786087017760,   2971460954327838336,   2666695728242931840,   1641043525072573440,   662729115894693120,   158601839701294080,   17080198121677824;
  9: 317475837322472439, 3760867611358519662, 20250825599622798180, 65425744244935194120, 140916987604475802720, 212459458234440441024, 228802493483243551872, 176001918064033501440, 94770263572941116160, 34020094615927580160, 7327404994199786496, 717368321110468608;

%F Formula

The row polynomials satisfy P(n,x) = (39+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1521 + 3276*x + 1764*x^2.

%t Mathematica

P[0, x_] := 1521 + 3276*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[(39 + 42*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1521 + 3276*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (39 + 42*x)*v[i - 1]); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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
    
    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

Related ?

Same fingerprint ?

A038327 → x ↦ x+1 → x ↦ x/(x+1)

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.