A322944 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	9	8	1													18
4	60	71	18	1												150
5	513	736	278	32	1											1560
6	5286	8805	4360	770	50	1										19272
7	63417	119496	72415	17280	1735	72	1									274416
8	865824	1812363	1288826	386995	53060	3409	98	1								4410576
9	13229505	30347136	24627708	8891456	1552390	136640	6076	128	1							78791040
10	223336458	555508521	504413568	212194052	45093636	5077758	309456	10068	162	1						1545943680
11	4123226601	11026937160	11040285645	5289418240	1325514050	182685552	14263410	635520	15765	200	1					33002982144
12	82559530692	235768997871	257356842930	138024237285	39842038720	6514765334	623933772	35613930	1208460	23595	242	1				760727192832
13	1780580892609	5399004554208	6367767149106	3771994527840	1231796685295	233486275968	26604604796	1863810432	80964015	2159520	34034	288	1			18813181658112
14	41125146159150	131770261208781	166702298286888	107905201512966	39301002485030	8481781574695	1123973648512	93781106276	4998750042	170525355	3666520	47606	338	1		496508618972160
15	1012187976013593	3413203637638536	4603665338850123	3228039164418432	1296258992229201	313985975399096	47542315207387	4620063475456	293336542499	12270810552	337089753	5963776	64883	392	1	13919809413703680

Technical details

Node id: n-2d6f253a90ee076c · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d887d9df7a78af17

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:         2,         1;
   3:         9,         8,         1;
   4:        60,        71,        18,         1;
   5:       513,       736,       278,        32,        1;
   6:      5286,      8805,      4360,       770,       50,       1;
   7:     63417,    119496,     72415,     17280,     1735,      72,      1;
   8:    865824,   1812363,   1288826,    386995,    53060,    3409,     98,     1;
   9:  13229505,  30347136,  24627708,   8891456,  1552390,  136640,   6076,   128,   1;
  10: 223336458, 555508521, 504413568, 212194052, 45093636, 5077758, 309456, 10068, 162, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) + (3+3*x)*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[(n + x)*P[n - 1, x] + (3 + 3*x)*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] = (n + x)*v[i - 1] + (3 + 3*x)*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 Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.