A128813 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	3	4	1													8
3	18	27	10	1												56
4	180	288	127	20	1											616
5	2700	4500	2193	427	35	1										9856
6	56700	97200	50553	11160	1162	56	1									216832
7	1587600	2778300	1512684	363033	43696	2730	84	1								6288128
8	57153600	101606400	57234924	14581872	1936089	141976	5754	120	1							232660736
9	2571912000	4629441600	2677177980	713419164	101705877	8325009	400906	11154	165	1						10702393856
10	141455160000	257191200000	151874230500	41915232000	6307242399	559581372	30374839	1014376	20229	220	1					599334055936
11	9336040560000	17116074360000	10280890413000	2918279542500	458193230334	43239612951	2564320746	97323655	2349490	34749	286	1				40155381747712
12	728211163680000	1344389840640000	819025526574000	237906694728000	38657351508552	3830883040512	243256631139	10155565836	280583875	5059912	57057	364	1			3172275158069248
13	66267215894880000	123067686661920000	75875712758874000	22468534746822000	3755725682006232	387267708195144	25967236474161	1167413122215	35688698461	741035867	10252099	90181	455	1		291849314542370816
14	6958057668962400000	12988374315396480000	8090017526343690000	2435071861175184000	416819731357476360	44418835042496352	3113827537982049	148545614306736	4914726460620	113497464496	1817506262	19721104	137956	560	1	30936027341491306496

Technical details

Node id: n-a571d223aed400cf · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-b22b54c97373a52a

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:          1;
  1:          1,          1;
  2:          3,          4,          1;
  3:         18,         27,         10,         1;
  4:        180,        288,        127,        20,         1;
  5:       2700,       4500,       2193,       427,        35,       1;
  6:      56700,      97200,      50553,     11160,      1162,      56,      1;
  7:    1587600,    2778300,    1512684,    363033,     43696,    2730,     84,     1;
  8:   57153600,  101606400,   57234924,  14581872,   1936089,  141976,   5754,   120,   1;
  9: 2571912000, 4629441600, 2677177980, 713419164, 101705877, 8325009, 400906, 11154, 165, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + 2*x)*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + n + 2*x)*v[i - 1])/(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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - 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 ?

A128813: Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).

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.