A142963 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	15	18	4													37
3	111	198	96	8												413
4	1071	2562	2004	528	16											6181
5	12663	37998	41136	18792	3024	32										113645
6	177255	639738	885060	577200	172320	17760	64									2469397
7	2866671	12093030	20385216	17326440	7642080	1579968	105600	128								61999133
8	52587711	253921554	505029492	530034192	311102640	98768448	14635392	631296	256							1766710981
9	1078805223	5867608734	13452958224	16818013800	12356943984	5336410464	1267053312	137509632	3782400	512						56319086285
10	24470577495	148040328618	384460528356	558029667312	493148585664	270391449888	89366526144	16272872448	1310781696	22682112	1024					1985514000757
11	608113846431	4050452914902	11752620804000	19430527617000	20082247056000	13376397355776	5696248102272	1479501475200	210167635200	12654144000	136065024	2048				76699067017853
12	16429992037551	119475628707906	383080760404884	710806343922000	841554672705360	660451714108416	345198040782336	117314432054784	24397855948800	2735798638080	123411876864	816328704	4096			3221569467519781
13	479502707791383	3779942702596110	13272609960246384	27314926182200808	36459622829633040	32966521563634848	20456730717143040	8624229381679104	2385061662233088	402629467115520	35929416425472	1212894074880	4897837056	8192		146178924382618925
14	15032820820737735	127694736183869658	487339807860099972	1101732074410564656	1637138892598003296	1676574945890177760	1205690415471183168	608205817910771712	210973906649304576	48172438339439616	6668363657226240	476151838175232	11986903154688	29386727424	16384	7125712387919452117

Technical details

Node id: n-c25403f40cbbc459 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-ff9c3889e2bd0aed

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:          3,          2;
  2:         15,         18,           4;
  3:        111,        198,          96,           8;
  4:       1071,       2562,        2004,         528,          16;
  5:      12663,      37998,       41136,       18792,        3024,         32;
  6:     177255,     639738,      885060,      577200,      172320,      17760,         64;
  7:    2866671,   12093030,    20385216,    17326440,     7642080,    1579968,     105600,       128;
  8:   52587711,  253921554,   505029492,   530034192,   311102640,   98768448,   14635392,    631296,     256;
  9: 1078805223, 5867608734, 13452958224, 16818013800, 12356943984, 5336410464, 1267053312, 137509632, 3782400, 512;

%F Formula

The row polynomials satisfy P(n,x) = (4*n-1+(4*n-2)*x)*P(n-1,x) - (3+7*x+4*x^2)*d/dx 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[(4*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (3 + 7*x + 4*x^2)*D[P[n - 1, x], x]];
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] = (4*n - 1 + (4*n - 2)*x)*v[i - 1] - (3 + 7*x + 4*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 - 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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