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	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

All 15 rows

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.