A159854 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 6\,x + 2\,x^{2} + n x + n x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 13\,x^{2} + 3\,x^{3} + 2\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 4\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	4	3															8
1	2	10	12	4														28
2	4	24	39	24	5													96
3	8	56	114	100	40	6												324
4	16	128	312	352	205	60	7											1080
5	32	288	816	1120	850	366	84	8										3564
6	64	640	2064	3328	3100	1752	595	112	9									11664
7	128	1408	5088	9408	10360	7224	3234	904	144	10								37908
8	256	3072	12288	25600	32480	26880	14896	5504	1305	180	11							122472
9	512	6656	29184	67584	96960	92736	61152	28032	8802	1810	220	12						393660
10	1024	14336	68352	174080	278400	301824	230496	125952	49140	13400	2431	264	13					1259712
11	2048	30720	158208	439296	774400	937728	813120	515328	239976	81400	19602	3180	312	14				4015332
12	4096	65536	362496	1089536	2097920	2804736	2720256	1959936	1059696	429440	128744	27744	4069	364	15			12754584
13	8192	139264	823296	2662400	5557760	8126976	8712704	7028736	4324320	2036320	729872	195936	38194	5110	420	16		40389516
14	16384	294912	1855488	6422528	14443520	22923264	26906880	24014848	16555968	8877440	3699696	1188096	288652	51352	6315	480	17	127545840

Technical details

Node id: n-98f16fdd4b44e5ec · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-31c7f065f4c4bbdd

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1,    4,     3;
  1:   2,   10,    12,     4;
  2:   4,   24,    39,    24,     5;
  3:   8,   56,   114,   100,    40,     6;
  4:  16,  128,   312,   352,   205,    60,     7;
  5:  32,  288,   816,  1120,   850,   366,    84,     8;
  6:  64,  640,  2064,  3328,  3100,  1752,   595,   112,    9;
  7: 128, 1408,  5088,  9408, 10360,  7224,  3234,   904,  144,   10;
  8: 256, 3072, 12288, 25600, 32480, 26880, 14896,  5504, 1305,  180,  11;
  9: 512, 6656, 29184, 67584, 96960, 92736, 61152, 28032, 8802, 1810, 220, 12;

%F Formula

The row polynomials satisfy (2+(n+6)*x+(n+2)*x^2)*P(n,x) = (4+(2*n+16)*x+(3*n+13)*x^2+(n+3)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x + 3*x^2.

%t Mathematica

P[0, x_] := 1 + 4*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (2*n + 16)*x + (3*n + 13)*x^2 + (n + 3)*x^3)*P[n - 1, x])/(2 + (n + 6)*x + (n + 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (2*n + 16)*x + (3*n + 13)*x^2 + (n + 3)*x^3)*v[i - 1])/(2 + (n + 6)*x + (n + 2)*x^2)); 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 shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)

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.