A390433 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	5	6	1													12
4	57	79	23	1												160
5	1169	1820	710	60	1											3760
6	37673	64469	30442	3770	125	1										136480
7	1749541	3237138	1757719	284012	14115	226	1									7042752
8	110331249	218042083	132142865	26123963	1733387	41825	371	1								488415744
9	9051479425	18932667832	12563623548	2917104136	242418182	7853704	105084	568	1							43715252480
10	935328762865	2056450561289	1474545224932	390909874964	39052606046	1594595326	28670612	233700	825	1						4897910530560
11	118718609669285	272913194149982	209392380648061	62093629605832	7245983390154	358186387156	8124912738	88949384	473265	1150	1					670730198187008
12	18140541415043529	43418739583412887	35378782409238075	11555039347519253	1541733130985930	89711076530806	2466862771366	34052831098	243143549	889955	1551	1				110127048122368000
13	3282534627872283089	8152127137556931412	7012270733874776290	2492356798595514916	373965949695755343	25087348847039656	813457839884700	13606067445480	122383309631	600651524	1575970	2036	1			21339169783335170048
14	693795240948153885785	1782760908433361632285	1610892094585046722350	617158431955675602958	102765422165362722307	7820694716545752743	292991971743285812	5777081094663060	62903556857167	388549174299	1366004926	2655614	2613	1		4815491625150458961920
15	169301519678057018099845	449032386279987158959978	424473293522516104626163	173942058727352471081940	31796542194458670376373	2709778060175717662566	115464454225642233419	2630261832892873176	33730944696427927	251964900726422	1113905082433	2899830804	4292015	3290	1	1251373707162632082276352

Technical details

Node id: n-a918183f5510afb3 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-e2bce5ba43dfc922

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:            5,             6,             1;
   4:           57,            79,            23,            1;
   5:         1169,          1820,           710,           60,           1;
   6:        37673,         64469,         30442,         3770,         125,          1;
   7:      1749541,       3237138,       1757719,       284012,       14115,        226,        1;
   8:    110331249,     218042083,     132142865,     26123963,     1733387,      41825,      371,      1;
   9:   9051479425,   18932667832,   12563623548,   2917104136,   242418182,    7853704,   105084,    568,   1;
  10: 935328762865, 2056450561289, 1474545224932, 390909874964, 39052606046, 1594595326, 28670612, 233700, 825, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n^2-4*n+5+x)*P(n-1,x) + (2*n-3+(2*n-3)*x)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^2 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^2 - 4*n + 5 + x)*P[n - 1, x] + (2*n - 3 + (2*n - 3)*x)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*D[P[n - 1, x], {x, 2}]];
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^2 - 4*n + 5 + x)*v[i - 1] + (2*n - 3 + (2*n - 3)*x)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*deriv(deriv(v[i - 1], x), 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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, Lah numbers with level 2.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.