A128813 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 - 4\,n + 4\,n x - n^{2} x^{2} - 4\,n^{3} x + n^{3} x^{2} + n^{4} x^{2} - n^{5} x^{2}\right)\,P_{n-2}(x) + \left(8\,n - 4\,n x + 4\,n^{3} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + x\)

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	5	10	3													18
3	11	64	81	18												174
4	21	274	991	1008	180											2474
5	36	889	7006	20193	18000	2700										48824
6	57	2380	34642	213372	536553	437400	56700									1281104
7	85	5544	133818	1495828	7926453	18182484	13891500	1587600								43223312
8	121	11628	431682	7886332	74845449	357991416	768479724	558835200	57153600							1825635152
9	166	22473	1213872	33700942	516854394	4382220861	19453665024	39712710780	27776649600	2571912000						94448950112
10	221	40678	3063357	122513732	2828281699	38403035766	299713866399	1256909076000	2466595030500	1671742800000	141455160000					5877772868352
11	287	69784	7083219	391644110	12918927385	262001998452	3250592225289	23804429570334	95446293259500	181441634013000	119812520520000	9336040560000				433366829871360
12	365	114478	15236793	1127395412	51058413055	1469695352670	27059437914723	313089695108928	2179817604060552	8428161960468000	15607313773614000	10082923804800000	728211163680000			37368099336158976
13	456	180817	30846478	2974395567	179184528172	7040167431751	182938068441342	3124108902035313	34188798846251232	228441073150226232	857101375094436000	1552687952701914000	984541493295360000	66267215894880000		3726542178310927360
14	561	276472	59301268	7289746152	569304828742	29601856228216	1044734026607772	25059165918163128	402883342920449217	4212616148617259952	27202610204284500360	99515282177299464000	176938883626497930000	110401181680870080000	6958057668962400000	425657648927907235840

Technical details

Node id: n-7a8f239d989ba059 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-cc8071de9b089857

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:   5,    10,       3;
  3:  11,    64,      81,       18;
  4:  21,   274,     991,     1008,       180;
  5:  36,   889,    7006,    20193,     18000,       2700;
  6:  57,  2380,   34642,   213372,    536553,     437400,       56700;
  7:  85,  5544,  133818,  1495828,   7926453,   18182484,    13891500,     1587600;
  8: 121, 11628,  431682,  7886332,  74845449,  357991416,   768479724,   558835200,    57153600;
  9: 166, 22473, 1213872, 33700942, 516854394, 4382220861, 19453665024, 39712710780, 27776649600, 2571912000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(4*n+4+(4*n^3-4*n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n-2,x) + (8*n+(4*n^3-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (8*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (8*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? 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.