A391533 → Pₙ + Pₙ₊₁ → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 12 + 12\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	12	12	3															27
1	20	36	21	4														81
2	28	76	75	32	5													216
3	36	132	189	132	45	6												540
4	44	204	387	384	210	60	7											1296
5	52	292	693	900	690	312	77	8										3024
6	60	396	1131	1824	1815	1140	441	96	9									6912
7	68	516	1725	3332	4095	3318	1771	600	117	10								15552
8	76	652	2499	5632	8260	8232	5642	2624	792	140	11							34560
9	84	804	3477	8964	15300	18144	15246	9072	3744	1020	165	12						76032
10	92	972	4683	13600	26505	36504	36414	26496	13950	5180	1287	192	13					165888
11	100	1156	6141	19844	43505	68310	79002	68112	43758	20680	6985	1596	221	14				359424
12	108	1356	7875	28032	68310	120516	158697	158400	120384	69300	29733	9216	1950	252	15			774144
13	116	1572	9909	38532	103350	202488	299299	339768	298584	203060	105963	41652	11934	2352	285	16		1658880
14	124	1804	12267	51744	151515	326508	535535	681824	680823	534820	329329	157248	57057	15204	2805	320	17	3538944

Technical details

Node id: n-c3f5414409369971 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-83046ed9368045bd

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: 12,  12,    3;
  1: 20,  36,   21,    4;
  2: 28,  76,   75,   32,     5;
  3: 36, 132,  189,  132,    45,     6;
  4: 44, 204,  387,  384,   210,    60,     7;
  5: 52, 292,  693,  900,   690,   312,    77,    8;
  6: 60, 396, 1131, 1824,  1815,  1140,   441,   96,    9;
  7: 68, 516, 1725, 3332,  4095,  3318,  1771,  600,  117,   10;
  8: 76, 652, 2499, 5632,  8260,  8232,  5642, 2624,  792,  140,  11;
  9: 84, 804, 3477, 8964, 15300, 18144, 15246, 9072, 3744, 1020, 165, 12;

%F Formula

The row polynomials satisfy (2*n+1+(n+2)*x)*P(n,x) = (2*n+3+(3*n+6)*x+(n+3)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 12*x + 3*x^2.

%t Mathematica

P[0, x_] := 12 + 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 3 + (3*n + 6)*x + (n + 3)*x^2)*P[n - 1, x])/(2*n + 1 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 12*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 3 + (3*n + 6)*x + (n + 3)*x^2)*v[i - 1])/(2*n + 1 + (n + 2)*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from shift_x_plus_one of Riordan array ((1+x)^3,x).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    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
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (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.