A142075 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

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	2														5
3	13	16	4													33
4	75	134	68	8												285
5	541	1268	984	272	16											3081
6	4683	13586	14248	6384	1072	32										40005
7	47293	163560	216444	135232	39216	4224	64									606033
8	545835	2191934	3505020	2818968	1178384	234912	16704	128								10491885
9	7087261	32410396	60785392	59898448	32907040	9796672	1391872	66304	256							204343641
10	102247563	524566250	1129081088	1318351808	900147296	359905856	79301120	8216576	263936	512						4422082005
11	1622632573	9228981584	22423353396	30284929920	24758721696	12485395968	3779445888	632297472	48503040	1052672	1024					105265315233
12	28091567595	175430841494	474921849652	728574493224	694477837536	423828596160	164295898752	38671029504	5000402688	286797312	4203520	2048				2733583519485
13	526858348381	3583832435844	10697270176584	18377717534480	20023409785200	14364155969664	6812974430976	2086231681536	389129521920	39387591680	1699743744	16797696	4096			76902684021801
14	10641342970443	78315755164994	255521499565080	486045220364688	596084717619280	491913401183712	276539643622656	104785995947520	25847680419072	3874297781760	309816727552	10097627136	67153920	8192		2329889536156005
15	230283190977853	1823124238332088	6455068463192620	13469081526378560	18366868041632336	17149244107221632	11162875329523392	5056140822190080	1561373665324800	314822593316864	38322087191552	2437337464832	60115578880	268533760	16384	75629701786875633

Technical details

Node id: n-cf84c1880a723950 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4358062586543316

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,         2;
   3:        13,        16,          4;
   4:        75,       134,         68,          8;
   5:       541,      1268,        984,        272,        16;
   6:      4683,     13586,      14248,       6384,      1072,        32;
   7:     47293,    163560,     216444,     135232,     39216,      4224,       64;
   8:    545835,   2191934,    3505020,    2818968,   1178384,    234912,    16704,     128;
   9:   7087261,  32410396,   60785392,   59898448,  32907040,   9796672,  1391872,   66304,    256;
  10: 102247563, 524566250, 1129081088, 1318351808, 900147296, 359905856, 79301120, 8216576, 263936, 512;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-1+(2*n-2)*x)*P(n-1,x) - (1+3*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[(2*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (1 + 3*x + 2*x^2)*D[P[n - 1, x], x]];
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] = (2*n - 1 + (2*n - 2)*x)*v[i - 1] - (1 + 3*x + 2*x^2)*deriv(v[i - 1], x)); 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A156365 → x ↦ x+1

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