A038284 → 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

\(\left(7 + 24\,x + 12\,n x\right)\,P_{n}(x) = \left(98 + 672\,x + 864\,x^{2} + 168\,n x + 288\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 1344\,x + 1728\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	196	1344	1728								3268
1	2744	28224	72576	55296							158840
2	38416	526848	2032128	3096576	1658880						7352848
3	537824	9219840	47416320	108380160	116121600	47775744					329451488
4	7529536	154893312	995742720	3034644480	4877107200	4013162496	1337720832				14420800576
5	105413504	2529924096	19516557312	74348789760	159318835200	196644962304	131096641536	36691771392			620252895104
6	1475789056	40478785536	364309069824	1665412890624	4460927385600	7341411926016	7341411926016	4109478395904	990677827584		26315583996160
7	20661046784	637540872192	6557563256832	34973670703104	112415370117120	231254475669504	308339300892672	258897138941952	124825406275584	26418075402240	1104339203177984

All 15 rows

Technical details

Node id: n-80f979e5986f600d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f565be5c3ba48d91

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:           196,            1344,             1728;
  1:          2744,           28224,            72576,             55296;
  2:         38416,          526848,          2032128,           3096576,           1658880;
  3:        537824,         9219840,         47416320,         108380160,         116121600,           47775744;
  4:       7529536,       154893312,        995742720,        3034644480,        4877107200,         4013162496,         1337720832;
  5:     105413504,      2529924096,      19516557312,       74348789760,      159318835200,       196644962304,       131096641536,        36691771392;
  6:    1475789056,     40478785536,     364309069824,     1665412890624,     4460927385600,      7341411926016,      7341411926016,      4109478395904,       990677827584;
  7:   20661046784,    637540872192,    6557563256832,    34973670703104,   112415370117120,    231254475669504,    308339300892672,    258897138941952,    124825406275584,     26418075402240;
  8:  289254654976,   9917302456320,  114757356994560,   699473414062080,  2623025302732800,   6475125318746112,  10791875531243520,  12081866483957760,   8737778439290880,   3698530556313600,    697437190619136;
  9: 4049565169664, 152726457827328, 1963625886351360, 13464863220695040, 57706556660121600, 166194883181150208, 332389766362300416, 465151859632373760, 448539293216931840, 284786852836147200, 107405327355346944, 18260173718028288;

%F Formula

The row polynomials satisfy (7+(12*n+24)*x)*P(n,x) = (98+(168*n+672)*x+(288*n+864)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 1344*x + 1728*x^2.

%t Mathematica

P[0, x_] := 196 + 1344*x + 1728*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (168*n + 672)*x + (288*n + 864)*x^2)*P[n - 1, x])/(7 + (12*n + 24)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 1344*x + 1728*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (168*n + 672)*x + (288*n + 864)*x^2)*v[i - 1])/(7 + (12*n + 24)*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)*8^(i-j)*6^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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

Related ?

Same fingerprint ?

A038271 → x ↦ x/(x+1) → θ+1

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.