A391533 → reverse → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 2\,x + n + 2\,n x\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} + n + 3\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	4	8															12
1	5	16	12														33
2	6	26	36	16													84
3	7	38	75	64	20												204
4	8	52	132	164	100	24											480
5	9	68	210	340	305	144	28										1104
6	10	86	312	620	730	510	196	32									2496
7	11	106	441	1036	1505	1386	791	256	36								5568
8	12	128	600	1624	2800	3192	2408	1160	324	40							12288
9	13	152	792	2424	4830	6552	6132	3912	1629	400	44						26880
10	14	178	1020	3480	7860	12348	13776	10920	6030	2210	484	48					58368
11	15	206	1287	4840	12210	21780	28182	26664	18315	8910	2915	576	52				125952
12	16	236	1596	6556	18260	36432	53592	58872	48312	29260	12716	3756	676	56			270336
13	17	268	1950	8684	26455	58344	96096	120120	114543	82940	44902	17628	4745	784	60		577536
14	18	302	2352	11284	37310	90090	164164	229944	249678	210210	136136	66612	23842	5894	900	64	1228800

Technical details

Node id: n-b7fc054862727c62 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-645712c1d17a2bff

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
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:  4,   8;
  1:  5,  16,  12;
  2:  6,  26,  36,   16;
  3:  7,  38,  75,   64,   20;
  4:  8,  52, 132,  164,  100,   24;
  5:  9,  68, 210,  340,  305,  144,   28;
  6: 10,  86, 312,  620,  730,  510,  196,   32;
  7: 11, 106, 441, 1036, 1505, 1386,  791,  256,   36;
  8: 12, 128, 600, 1624, 2800, 3192, 2408, 1160,  324,  40;
  9: 13, 152, 792, 2424, 4830, 6552, 6132, 3912, 1629, 400, 44;

%F Formula

The row polynomials satisfy (n+3+(2*n+2)*x)*P(n,x) = (n+4+(3*n+8)*x+(2*n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (3*n + 8)*x + (2*n + 4)*x^2)*P[n - 1, x])/(n + 3 + (2*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] = 4 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (3*n + 8)*x + (2*n + 4)*x^2)*v[i - 1])/(n + 3 + (2*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 reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    no recorded property covers this requirement
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

A159854 → basis A013609 → D

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.