A269951 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

\(P_{2}(x) = 1 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	3	4	1													8
4	11	18	8	1												38
5	49	93	56	13	1											212
6	261	553	405	131	19	1										1370
7	1631	3760	3164	1270	260	26	1									10112
8	11743	28946	27047	12669	3255	463	34	1								84158
9	95901	249697	253385	133448	40404	7266	763	43	1							780908
10	876809	2390249	2594218	1499182	514703	109823	14658	1186	53	1						8000882
11	8877691	25171972	28899567	18017636	6837683	1664628	264957	27360	1761	64	1					89763320
12	98641011	289429114	348618814	231618561	95437505	25824812	4712190	582603	48015	2520	76	1				1094915222
13	1193556233	3609499773	4532478026	3179483747	1403929296	414513781	84433712	12012429	1189188	80135	3498	89	1			14431179908
14	15624736141	48542832801	63229385183	46497446313	21784681336	6923049298	1544751659	246199085	28132896	2283358	128271	4733	103	1		204423631178
15	220048367319	700453230880	942593874234	722537274914	356420514801	120656255434	29081119639	5093588786	653922269	61421646	4165447	198198	6266	118	1	3097603939952

Technical details

Node id: n-bc392a6ff7c9acaf · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-1710876ba539d99f

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 -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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      1,       1;
   3:      3,       4,       1;
   4:     11,      18,       8,       1;
   5:     49,      93,      56,      13,      1;
   6:    261,     553,     405,     131,     19,      1;
   7:   1631,    3760,    3164,    1270,    260,     26,     1;
   8:  11743,   28946,   27047,   12669,   3255,    463,    34,    1;
   9:  95901,  249697,  253385,  133448,  40404,   7266,   763,   43,  1;
  10: 876809, 2390249, 2594218, 1499182, 514703, 109823, 14658, 1186, 53, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>…?
  - ↑ Liu–Wang theorem for three-term recurrences (above)

References

[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