A038321 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + 14\,x + 7\,n x\right)\,P_{n}(x) = \left(108 + 504\,x + 441\,x^{2} + 126\,n x + 147\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 324 + 1512\,x + 1323\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	324	1512	1323															3159
1	5832	40824	71442	37044														155142
2	104976	979776	2571912	2667168	972405													7296237
3	1889568	22044960	77157360	120022560	87516450	24504606												333135504
4	34012224	476171136	2083248720	4320812160	4725888300	2646497448	600362847											14886992835
5	612220032	9999593856	52497867744	136105583040	198487308600	166729339224	75645718722	14408708328										654486339546
6	11019960576	205705930752	1259948825856	3919840791552	7145543109600	8003008282752	5446491747984	2074853999232	340405734249									28406818382553
7	198359290368	4165545097728	29158815684096	105835701371904	231515596751040	324121835451456	294110554391136	168063173937792	55145728948338	7942800465810								1220258111389668
8	3570467226624	83310901954560	656073352892160	2721489463848960	6945467902531200	11668386076252416	13234974947601120	10083790436267520	4963115605350420	1429704083845800	183478690760211							51973361928530991
9	64268410079232	1649555858700288	14433613763627520	67356864230261760	196457520671596800	385056740516329728	524105007925004352	499147626595242240	327565629953127720	141540704300734200	36328780770521778	4203330006506652						2197909643001732270
10	1156831381426176	32391278679932928	311766057294354432	1616564741526282240	5304353058133113600	11881750850218174464	18867780285300156672	21563177468914464768	17688544017468896880	10190930709652862400	3923508323216352024	907919281405436832	95625757648026333					92385468660839479749
11	20822964865671168	631629934258692096	6632114309716267008	37827614951715004416	137913179511460953600	347541212368881603072	630722940965748094464	840963921287664125952	827823860017544373984	596169446514692450400	306033649210875457872	106226555924436109344	22376427289638161922	2162613288347672454				3863045988539844637752
12	374813367582081024	12243903341014646784	139274400504041607168	866596269802925555712	3475412123688816030720	9731153946328684886016	19867772640421064975616	30274701166355908534272	34768602120736863707328	30046940104340499500160	19280119900285153845936	8923030697652633184896	2819429838494408402172	544978548663613458408	48658798987822630215			160799289272971033046427
13	6746640616477458432	236132421576711045120	2892622164314710302720	19498416070565825003520	85305570308725484390400	262741156550874491922432	596033179212631949268480	1021771164364511913031680	1341074653228421885854080	1352112304695322477507200	1041126474615398307680544	602304572091552739980480	253748685464496756195480	73572104069587816885080	13137875726712110158050	1089957097327226916816		6666651614722636883600514
14	121439531096594251776	4533742494272852066304	59505370237331183370240	431964909870996738539520	2047333687409411625369600	6879041189695623061241856	17165755561323800138932224	32696677259664381217013760	48278687516223187890746880	55629763393178981931724800	49974070781539118768666112	34692743352473437822875648	18269905353443766446074560	7062921990680430420967680	1891854104646543862759200	313907644030241352043008	24319667734113750581457	275423106964176735657224625

Technical details

Node id: n-21d0606f3943bc5f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-95fa3cdbaa55b585

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,             1512,              1323;
  1:           5832,            40824,             71442,             37044;
  2:         104976,           979776,           2571912,           2667168,             972405;
  3:        1889568,         22044960,          77157360,         120022560,           87516450,           24504606;
  4:       34012224,        476171136,        2083248720,        4320812160,         4725888300,         2646497448,          600362847;
  5:      612220032,       9999593856,       52497867744,      136105583040,       198487308600,       166729339224,        75645718722,        14408708328;
  6:    11019960576,     205705930752,     1259948825856,     3919840791552,      7145543109600,      8003008282752,      5446491747984,      2074853999232,       340405734249;
  7:   198359290368,    4165545097728,    29158815684096,   105835701371904,    231515596751040,    324121835451456,    294110554391136,    168063173937792,     55145728948338,      7942800465810;
  8:  3570467226624,   83310901954560,   656073352892160,  2721489463848960,   6945467902531200,  11668386076252416,  13234974947601120,  10083790436267520,   4963115605350420,   1429704083845800,   183478690760211;
  9: 64268410079232, 1649555858700288, 14433613763627520, 67356864230261760, 196457520671596800, 385056740516329728, 524105007925004352, 499147626595242240, 327565629953127720, 141540704300734200, 36328780770521778, 4203330006506652;

%F Formula

The row polynomials satisfy (6+(7*n+14)*x)*P(n,x) = (108+(126*n+504)*x+(147*n+441)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1512*x + 1323*x^2.

%t Mathematica

P[0, x_] := 324 + 1512*x + 1323*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((108 + (126*n + 504)*x + (147*n + 441)*x^2)*P[n - 1, x])/(6 + (7*n + 14)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1512*x + 1323*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((108 + (126*n + 504)*x + (147*n + 441)*x^2)*v[i - 1])/(6 + (7*n + 14)*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_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

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.