A160562 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(8 + 8\,x\right)\,P_{n-1}'(x) + \left(4 + 8\,x + 4\,x^{2}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	12	12	1													25
3	128	164	38	1												331
4	1872	3008	1224	88	1											6193
5	37600	73232	41216	5752	170	1										157971
6	990784	2226880	1567152	350656	19892	292	1									5155657
7	32333824	81824192	69174880	21669488	2040320	55804	462	1								207098971
8	1272660224	3584239616	3538509568	1418274560	200191968	8981632	134736	688	1							10022992993
9	59527313920	184385952000	207310471168	100795340032	19646510656	1334870624	32156672	290736	978	1						573032906787
10	3252626013184	10956410072064	13774755196160	7868620486656	1982642139264	191810691968	6915692448	98364032	575004	1340	1					38033879232121
11	204354574172160	742455784690688	1030807542896128	676328828712192	209637297393664	27391791141760	1406165909568	29440751648	265983872	1060884	1782	1				2892411692714347
12	14594815769038848	56846645682847744	86301717842163712	63872166336235520	23513685352492800	3961357644920832	278040350801664	8196265059072	107264168880	651449920	1849496	2312	1			249376733161030801
13	1174376539738169344	4877648235616407552	8028144182707716096	6606570371366684672	2817630736321011200	588251911593637632	54465180358189056	2185503126195456	39725679400992	344569654000	1471165696	3076008	2938	1		24149312732551310643
14	105595092426069327872	465409161655349592064	824528739067826860032	746073549927622393856	361838494711399816192	90637500419104087040	10712948350140218112	568827670792022016	13929760907925696	165638180149312	997804018640	3104835136	4916548	3668	1	2605378410628306166185

Technical details

Node id: n-4e4a24ae75ada18d · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4517b0bcdfb0f1f3

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:           1;
  1:           2,            1;
  2:          12,           12,            1;
  3:         128,          164,           38,            1;
  4:        1872,         3008,         1224,           88,           1;
  5:       37600,        73232,        41216,         5752,         170,          1;
  6:      990784,      2226880,      1567152,       350656,       19892,        292,        1;
  7:    32333824,     81824192,     69174880,     21669488,     2040320,      55804,      462,      1;
  8:  1272660224,   3584239616,   3538509568,   1418274560,   200191968,    8981632,   134736,    688,   1;
  9: 59527313920, 184385952000, 207310471168, 100795340032, 19646510656, 1334870624, 32156672, 290736, 978, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (8+8*x)*d/dx P(n-1,x) + (4+8*x+4*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (8 + 8*x)*D[P[n - 1, x], x] + (4 + 8*x + 4*x^2)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] + (8 + 8*x)*deriv(v[i - 1], x) + (4 + 8*x + 4*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 of scaled central factorial numbers, T(n,k) = A008958(n,n-k).?
  - 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.