A144696 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

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	12	15	4													31
3	60	108	57	8												233
4	360	840	660	195	16											2071
5	2520	7200	7500	3420	633	32										21305
6	20160	68040	88200	54600	16212	1995	64									249271
7	181440	705600	1093680	856800	354060	72828	6177	128								3270713
8	1814400	7983360	14394240	13653360	7265160	2134440	316020	18915	256							47580151
9	19958400	97977600	201398400	224380800	146429640	56397600	12252300	1339740	57513	512						760192505
10	239500800	1297296000	2993760000	3832012800	2966483520	1419790680	411899400	67960200	5589012	174075	1024					13234467511
11	3113510400	18441561600	47201616000	68257728000	61186466880	35090193600	12821830560	2879157600	367746060	23052348	525297	2048				249383390393
12	43589145600	280215936000	787718131200	1270052784000	1294441948800	866489944320	383296193280	109906917120	19481421960	1953992040	94310580	1582035	4096			5057242311031
13	653837184000	4533271142400	13883142873600	24693250982400	28209442060800	21609318931200	11248075238400	3945624883200	905718373560	128612484000	10241513100	383570460	4758393	8192		109820924003705
14	10461394944000	77806624896000	257829796224000	501493120128000	634832316518400	548006370912000	328573941696000	136743301094400	38810746054080	7240316002920	833123982600	53125417800	1553315412	14299755	16384	2542685745501751

Technical details

Node id: n-27283f27ccc0227a · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-51cb495c8cc1c7c8

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:       12,       15,         4;
  3:       60,      108,        57,         8;
  4:      360,      840,       660,       195,        16;
  5:     2520,     7200,      7500,      3420,       633,       32;
  6:    20160,    68040,     88200,     54600,     16212,     1995,       64;
  7:   181440,   705600,   1093680,    856800,    354060,    72828,     6177,     128;
  8:  1814400,  7983360,  14394240,  13653360,   7265160,  2134440,   316020,   18915,   256;
  9: 19958400, 97977600, 201398400, 224380800, 146429640, 56397600, 12252300, 1339740, 57513, 512;

%F Formula

The row polynomials satisfy P(n,x) = (n+2+(n+1)*x)*P(n-1,x) - (x+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[(n + 2 + (n + 1)*x)*P[n - 1, x] - (x + 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] = (n + 2 + (n + 1)*x)*v[i - 1] - (x + 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 of 2-Eulerian numbers.?
  - 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