A200536 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 14\,x - 72\,x^{2} - 42\,x^{3} + 36\,x^{4} + 42\,n x + 96\,n x^{2} - 42\,n x^{3} - 144\,n x^{4} + 49\,n^{2} x^{2} + 168\,n^{2} x^{3} + 144\,n^{2} x^{4}\right)\,P_{n}(x) = \left(24 + 252\,x + 1098\,x^{2} + 2541\,x^{3} + 3294\,x^{4} + 2268\,x^{5} + 648\,x^{6} + 252\,n x + 2046\,n x^{2} + 6594\,n x^{3} + 10530\,n x^{4} + 8316\,n x^{5} + 2592\,n x^{6} + 294\,n^{2} x^{2} + 2037\,n^{2} x^{3} + 5274\,n^{2} x^{4} + 6048\,n^{2} x^{5} + 2592\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	6	84	162																											252
2	36	1008	5913	12096	8100																									27153
3	216	9072	88938	365904	741150	734832	285768																							2225880
4	1296	72576	997272	6168960	20900025	41640480	48866328	31352832	8503056																					158502825
5	7776	544320	9622800	79470720	377682750	1131702516	2220774570	2860945920	2338340400	1102248000	228637728																			10349977500
6	46656	3919104	84715632	875266560	5354869500	21429905616	58997125089	114292829952	156147994620	147701232000	92255323248	34284321792	5748065856																	637175615625
7	279936	27433728	701029728	8696491776	65084682600	325958983056	1155440682018	2989720819008	5730042565926	8148974576400	8505266322168	6339742504704	3198798648864	979960197888	137749507200															37586164725000
8	1679616	188116992	5547771648	80325955584	710913218400	4276122617088	18545825959920	60061099650048	148221684433809	281536404609600	412171519803120	461821242645504	389267641867104	239110288284672	101108138284800	26330359136256	3184768606464													2146432072640625
9	10077696	1269789696	42447255552	702616965120	7178563310400	50407952628096	258581812243104	1005888033690624	3037474464607926	7221221266562100	13612385563613298	20369232682235136	24079771617658848	22229907108990336	15699517959844800	8195324281159680	2980943415650304	674818305831936	71607703822848											119495007075937500
10	60466176	8465264640	316313683200	5891824189440	68248361808000	547983457362432	3245577899669280	14740912393482240	52653592892273940	150367266518238000	346839859146702537	649586591358788160	988717466532699540	1218889193536062720	1207156269825972000	946915414322282496	575145324562849920	260929744921681920	83243955694060800	16662180390912000	1574576046941184									6517290374525390625
11	362797056	55870746624	2307661373952	47769488363520	618631336224000	5590350084920832	37544463425544768	195002510244562944	804279871147741560	2681884909235070000	7317668005989786522	16463169129856106448	30661633711048113774	47308449798906634800	60320990336080617000	63180813319238393856	53808878224198620864	36678286907165578752	19536550814728062720	7835390328826368000	2225663242351363584	399192517805469696	33997988931913728							349495655193984375000
12	2176782336	365699432448	16544634144768	376792315232256	5394607422969600	54184176395661312	406882934022571776	2378697260999786496	11125619659775004912	42427455419128348800	133678895670463343112	351174695181163491456	773688127045489322121	1433963338656417590112	2237188130022217105800	2932585718570151469056	3215304081674976419568	2926243636766174850048	2185285201301759014656	1316675486414569881600	624350500489106544384	224346195006673969152	57422603306002286592	9328688259820683264	723019613391360000					18478356945740947265625
13	13060694016	2377046310912	116899741790208	2904750591934464	45558374088614400	503498075611474944	4180383729327117312	27166294829564706816	142081874319168340128	609907584159741067200	2179310567074559311536	6547635423050631960960	16657875494803180623726	36050757288251177056788	66532934668592585331450	104762166768810111375360	140538515329609804197648	160064146386882445675968	153874669887659312358624	123776430817204195430400	82282492944345650052096	44413061751612593335296	18974711097259933496832	6174814237312973930496	1438086011035415040000	213490917779437928448	15179941387074281472			965280167750199023437500
14	78364164096	15359376162816	815829721362432	21963907912826880	374507360419046400	4516316085081894912	41081255141703678720	293771661115642085376	1698908170117931188416	8107101696055429267200	32394038847639569532432	109559370465455415418368	316108132638876842561388	782424662106732110160912	1667475972079166185167825	3065827247438623778589696	4865072502921412233740652	6655731755776416486665856	7828380908626683080147472	7880102848565877247718400	6742966527198068886823104	4858744585378063632924672	2909868302590117046680320	1422314392042671000846336	552857128134606771840000	164388006690167204904960	35118794398996350185472	4799610109677741539328	315218042136530141184	49904595780452439697265625

Technical details

Node id: n-ff998881405c954e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d4629d03e97d7675

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        6,         84,         162;
  2:       36,       1008,        5913,        12096,          8100;
  3:      216,       9072,       88938,       365904,        741150,         734832,          285768;
  4:     1296,      72576,      997272,      6168960,      20900025,       41640480,        48866328,         31352832,          8503056;
  5:     7776,     544320,     9622800,     79470720,     377682750,     1131702516,      2220774570,       2860945920,       2338340400,       1102248000,         228637728;
  6:    46656,    3919104,    84715632,    875266560,    5354869500,    21429905616,     58997125089,     114292829952,     156147994620,     147701232000,       92255323248,       34284321792,        5748065856;
  7:   279936,   27433728,   701029728,   8696491776,   65084682600,   325958983056,   1155440682018,    2989720819008,    5730042565926,    8148974576400,     8505266322168,     6339742504704,     3198798648864,      979960197888,      137749507200;
  8:  1679616,  188116992,  5547771648,  80325955584,  710913218400,  4276122617088,  18545825959920,   60061099650048,  148221684433809,  281536404609600,   412171519803120,   461821242645504,   389267641867104,   239110288284672,   101108138284800,   26330359136256,    3184768606464;
  9: 10077696, 1269789696, 42447255552, 702616965120, 7178563310400, 50407952628096, 258581812243104, 1005888033690624, 3037474464607926, 7221221266562100, 13612385563613298, 20369232682235136, 24079771617658848, 22229907108990336, 15699517959844800, 8195324281159680, 2980943415650304, 674818305831936, 71607703822848;

%F Formula

The row polynomials satisfy (4+(42*n-14)*x+(49*n^2+96*n-72)*x^2+(168*n^2-42*n-42)*x^3+(144*n^2-144*n+36)*x^4)*P(n,x) = (24+(252*n+252)*x+(294*n^2+2046*n+1098)*x^2+(2037*n^2+6594*n+2541)*x^3+(5274*n^2+10530*n+3294)*x^4+(6048*n^2+8316*n+2268)*x^5+(2592*n^2+2592*n+648)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((24 + (252*n + 252)*x + (294*n^2 + 2046*n + 1098)*x^2 + (2037*n^2 + 6594*n + 2541)*x^3 + (5274*n^2 + 10530*n + 3294)*x^4 + (6048*n^2 + 8316*n + 2268)*x^5 + (2592*n^2 + 2592*n + 648)*x^6)*P[n - 1, x])/(4 + (42*n - 14)*x + (49*n^2 + 96*n - 72)*x^2 + (168*n^2 - 42*n - 42)*x^3 + (144*n^2 - 144*n + 36)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((24 + (252*n + 252)*x + (294*n^2 + 2046*n + 1098)*x^2 + (2037*n^2 + 6594*n + 2541)*x^3 + (5274*n^2 + 10530*n + 3294)*x^4 + (6048*n^2 + 8316*n + 2268)*x^5 + (2592*n^2 + 2592*n + 648)*x^6)*v[i - 1])/(4 + (42*n - 14)*x + (49*n^2 + 96*n - 72)*x^2 + (168*n^2 - 42*n - 42)*x^3 + (144*n^2 - 144*n + 36)*x^4)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.