A038321 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 28\,x + 14\,n x\right)\,P_{n}(x) = \left(162 + 1008\,x + 1176\,x^{2} + 252\,n x + 392\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 324 + 2016\,x + 2352\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	324	2016	2352															4692
1	5832	54432	127008	87808														275080
2	104976	1306368	4572288	6322176	3073280													15379088
3	1889568	29393280	137168640	284497920	276595200	103262208												832806816
4	34012224	634894848	3703553280	10241925120	14936140800	11152318464	3373232128											44076076864
5	612220032	13332791808	93329542656	322620641280	627317913600	702596063232	425027248128	107943428096										2292779848832
6	11019960576	274274574336	2239909023744	9291474468864	22583444889600	33724611035136	30601961865216	15543853645824	3400217985024									117670767448320
7	198359290368	5554060130304	51837894549504	250869810659328	731703614423040	1365846746923008	1652505940721664	1259052145311744	550835313573888	105784559534080								5974188445116928
8	3570467226624	111081202606080	1166352627363840	6450937988382720	21951108432691200	49170482889228288	74362767332474880	75543128718704640	49575178221649920	19041220716134400	3258164433649664							300633993030112256
9	64268410079232	2199407811600384	25659757802004480	159660715212472320	620902781381836800	1622625935344533504	2944765586366005248	3739384871575879680	3271961762628894720	1885080850897305600	645116557862633472	99522113609662464						15016944608902907904
10	1156831381426176	43188371573243904	554250768523296768	3831857165099335680	16764375097309593600	50069600290631319552	106011561109176188928	161541426452078002176	176685935181960314880	135725821264606003200	69672588249164414976	21496776539687092224	3018837446159761408					745417374767349993472
11	20822964865671168	842173245678256128	11790425439495585792	89665457663324454912	435873752530049433600	1464535808500966096896	3543815042792461172736	6300115631631042084864	8268901766515742736384	7939960543979451187200	5434461883434824368128	2515122855143389790208	706407962401384169472	91029559914971267072				36802543686157646274560
12	374813367582081024	16325204454686195712	247598934229407301632	2054154121014342057984	10984018563757245726720	41007002638027050713088	111630173847962526941184	226804162738717515055104	347293874193661194928128	400174011416564339834880	342371098656393935192064	211270319832044742377472	89007403262574405353472	22939449098572759302144	2730886797449138012160			1808530854118790871072768
13	6746640616477458432	314843228768948060160	5142439403226151649280	46218467722822696304640	269607728383132395110400	1107189071226730369253376	3348905215438875808235520	7654640492431716133109760	13395620861755503232942080	18007830513745395292569600	18488039327445272500371456	14260746588663020110479360	8010666293631696481812480	3096825628307322505789440	737339435311267263283200	81562485683814255296512		88510656139019180621725696
14	121439531096594251776	6044989992363802755072	105787324866366548213760	1023916823397918195056640	6470585481195177482649600	28988222955754395122270208	96448470204639623277182976	244948495757814916259512320	482242351023198116385914880	740893598279810549180006400	887425887717373080017829888	821419003506989958363611136	576767973141482146690498560	297295260317502960555786240	106176878684822485912780800	23489995876938505525395456	2426483949093474095071232	4316129077474403134008786944

Technical details

Node id: n-34511d415dd7604c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c7671229903995ff

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:            324,             2016,              2352;
  1:           5832,            54432,            127008,              87808;
  2:         104976,          1306368,           4572288,            6322176,            3073280;
  3:        1889568,         29393280,         137168640,          284497920,          276595200,           103262208;
  4:       34012224,        634894848,        3703553280,        10241925120,        14936140800,         11152318464,          3373232128;
  5:      612220032,      13332791808,       93329542656,       322620641280,       627317913600,        702596063232,        425027248128,        107943428096;
  6:    11019960576,     274274574336,     2239909023744,      9291474468864,     22583444889600,      33724611035136,      30601961865216,      15543853645824,       3400217985024;
  7:   198359290368,    5554060130304,    51837894549504,    250869810659328,    731703614423040,    1365846746923008,    1652505940721664,    1259052145311744,     550835313573888,     105784559534080;
  8:  3570467226624,  111081202606080,  1166352627363840,   6450937988382720,  21951108432691200,   49170482889228288,   74362767332474880,   75543128718704640,   49575178221649920,   19041220716134400,   3258164433649664;
  9: 64268410079232, 2199407811600384, 25659757802004480, 159660715212472320, 620902781381836800, 1622625935344533504, 2944765586366005248, 3739384871575879680, 3271961762628894720, 1885080850897305600, 645116557862633472, 99522113609662464;

%F Formula

The row polynomials satisfy (9+(14*n+28)*x)*P(n,x) = (162+(252*n+1008)*x+(392*n+1176)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 2016*x + 2352*x^2.

%t Mathematica

P[0, x_] := 324 + 2016*x + 2352*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((162 + (252*n + 1008)*x + (392*n + 1176)*x^2)*P[n - 1, x])/(9 + (14*n + 28)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 2016*x + 2352*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((162 + (252*n + 1008)*x + (392*n + 1176)*x^2)*v[i - 1])/(9 + (14*n + 28)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013611 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

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.