A056242 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 5\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	1														4
3	10	7	1													18
4	34	37	12	1												84
5	116	172	89	18	1											396
6	396	744	531	175	25	1										1872
7	1352	3076	2802	1287	305	33	1									8856
8	4616	12332	13670	8065	2688	490	42	1								41904
9	15760	48352	63160	45544	19609	5068	742	52	1							198288
10	53808	186416	280392	239024	126283	42381	8862	1074	63	1						938304
11	183712	709296	1207264	1187960	744170	307583	83913	14622	1500	75	1					4440096
12	627232	2670352	5073808	5660728	4102570	2016037	679668	155199	23034	2035	88	1				21010752
13	2141504	9966144	20909808	26083520	21472252	12233652	4916001	1391566	271755	34936	2695	102	1			99423936
14	7311552	36925184	84785616	116940464	107794916	69878600	32621291	11032827	2678819	454883	51337	3497	117	1		470479104
15	24963200	135967040	339133280	512470480	522955720	380136652	202237154	79609103	23146721	4900467	733161	73437	4459	133	1	2226331008

Technical details

Node id: n-f86d2c725a2a99ea · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-869347430af03884

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:     3,      1;
   3:    10,      7,      1;
   4:    34,     37,     12,      1;
   5:   116,    172,     89,     18,      1;
   6:   396,    744,    531,    175,     25,     1;
   7:  1352,   3076,   2802,   1287,    305,    33,    1;
   8:  4616,  12332,  13670,   8065,   2688,   490,   42,    1;
   9: 15760,  48352,  63160,  45544,  19609,  5068,  742,   52,  1;
  10: 53808, 186416, 280392, 239024, 126283, 42381, 8862, 1074, 63, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n-7+(3*n-5)*x)*P(n-1,x) - (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 7 + (3*n - 5)*x)*P[n - 1, x] - (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((5*n - 7 + (3*n - 5)*x)*v[i - 1] - (4 + 6*x + 2*x^2)*deriv(v[i - 1], x))/(n - 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 Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    ↑ Affine-current basis map theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A056242 → x ↦ x/(x+1)