A038330 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 36\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 784 + 2016\,x + 1296\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	784	2016	1296															4096
1	21952	84672	108864	46656														262144
2	614656	3161088	6096384	5225472	1679616													16777216
3	17210368	110638080	284497920	365783040	235146240	60466176												1073741824
4	481890304	3717439488	11948912640	20483850240	19752284160	10158317568	2176782336											68719476736
5	13492928512	121436356608	468397375488	1003708661760	1290482565120	995515121664	426649337856	78364164096										4398046511104
6	377801998336	3885963411456	17486835351552	44966148046848	72267023646720	74331795750912	47784725839872	17553572757504	2821109907456									281474976710656
7	10578455953408	122407847460864	629526072655872	1888578217967616	3642257991794688	4682903132307456	4013916970549248	2211750167445504	710919696678912	101559956668416								18014398509481984
8	296196766695424	3808244143226880	22033412542955520	75543128718704640	169972039617085440	262242575409217536	280974187938447360	206430015628247040	99528757535047680	28436787867156480	3656158440062976							1152921504606846976
9	8293509467471872	117293919611387904	754032340358922240	2908410455670128640	7478769743151759360	13461785537673166848	17308009977008357376	15895111203375022080	10218285773598228480	4379265331542097920	1126096799539396608	131621703842267136						73786976294838206464
10	232218265089212416	3582796089947848704	25335486636059787264	108580657011684802560	314108329212373893120	646165705808312008704	969248558712468013056	1068151472866801483776	858336004982251192320	490477717132714967040	189184262322618630144	44224892491001757696	4738381338321616896					4722366482869645213696
11	6502111422497947648	108678148061751410688	838374285047796596736	3952335915225326813184	12703936870367121899520	29400539614278196396032	50400925053048336678912	64801189353919290015744	62486861162707886800896	44633472259077062000640	22954357161811060457472	8048930433362319900672	1724770807149068550144	170581728179578208256				302231454903657293676544
12	182059119829942534144	3277064156938965614592	27386893311561355493376	140846879888029828251648	497994325318391178461184	1280556836533005887471616	2469645327599368497266688	3628866603819480240881664	4082474929296915270991872	3499264225111641660850176	2249527001857483924832256	1051726909959343133687808	338055078201217435828224	66868037446394657636352	6140942214464815497216			19342813113834066795298816
13	5097655355238390956032	98311924708168968437760	884807322373520715939840	4929640796081043988807680	19014328784884026813972480	53783387134386247273807872	115250115287970529872445440	190515496700522712646287360	244948495757814916259512320	244948495757814916259512320	188960268156028649685909504	110431325545731029037219840	47327710948170441015951360	14042287863742878103633920	2579195730075222508830720	221073919720733357899776		1237940039285380274899124224
14	142734349946674946768896	2936249484617313190674432	28313834315952662910074880	169883005895715977460449280	709868274635670334388305920	2190450676018639888969629696	5163205164901079738285555712	9483438057981575029504081920	13717115762437635310532689920	15676703728500154640608788480	14109033355650139176547909632	9894646768897500201734897664	5300703626195089393786552320	2096981654318936463475998720	577739843536849841978081280	99041116034888544339099648	7958661109946400884391936	79228162514264337593543950336

Technical details

Node id: n-b0181b55e9572fd7 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-27210632b0078bca

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:              784,               2016,               1296;
  1:            21952,              84672,             108864,               46656;
  2:           614656,            3161088,            6096384,             5225472,             1679616;
  3:         17210368,          110638080,          284497920,           365783040,           235146240,             60466176;
  4:        481890304,         3717439488,        11948912640,         20483850240,         19752284160,          10158317568,           2176782336;
  5:      13492928512,       121436356608,       468397375488,       1003708661760,       1290482565120,         995515121664,         426649337856,          78364164096;
  6:     377801998336,      3885963411456,     17486835351552,      44966148046848,      72267023646720,       74331795750912,       47784725839872,       17553572757504,        2821109907456;
  7:   10578455953408,    122407847460864,    629526072655872,    1888578217967616,    3642257991794688,     4682903132307456,     4013916970549248,     2211750167445504,      710919696678912,     101559956668416;
  8:  296196766695424,   3808244143226880,  22033412542955520,   75543128718704640,  169972039617085440,   262242575409217536,   280974187938447360,   206430015628247040,    99528757535047680,   28436787867156480,    3656158440062976;
  9: 8293509467471872, 117293919611387904, 754032340358922240, 2908410455670128640, 7478769743151759360, 13461785537673166848, 17308009977008357376, 15895111203375022080, 10218285773598228480, 4379265331542097920, 1126096799539396608, 131621703842267136;

%F Formula

The row polynomials satisfy P(n,x) = (28+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 2016*x + 1296*x^2.

%t Mathematica

P[0, x_] := 784 + 2016*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 36*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] = 784 + 2016*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 36*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)*12^(i-j)*4^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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

Related ?

Same fingerprint ?

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.