A144697 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 1 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	6														7
2	1	20	27													48
3	1	50	201	108												360
4	1	112	978	1504	405											3000
5	1	238	3942	12536	9545	1458										27720
6	1	492	14325	81680	126075	54564	5103									282240
7	1	1002	48879	460420	1242975	1083834	290353	17496								3144960
8	1	2024	160164	2363056	10243550	15452424	8354836	1467296	59049							38102400
9	1	4070	510852	11374192	74848310	178760364	166650820	59480288	7134273	196830						498960000
10	1	8164	1599711	52308128	502201210	1791105960	2626432046	1618590784	398785077	33676180	649539					7025356800
11	1	16354	4946085	232694940	3164857650	16170408828	35102013282	33960182640	14514553485	2552234450	155320121	2125764				105859353600
12	1	32736	15157158	1009917120	19023843675	135000744768	415503385668	592614153984	398067419655	122286034400	15745810806	703269696	6908733			1699976678400
13	1	65502	46158930	4302392552	110267975175	1061119205118	4483532388636	9014662341600	8916134162751	4318940274650	980291745906	94325477112	3137315545	22320522		28986781824000
14	1	131036	139950585	18071203568	621253754845	7954574073228	45002423186133	123357668888640	171518663673459	122443928637620	44039049319771	7548360789456	551714769775	13827077348	71744535	523069747200000

Technical details

Node id: n-f31158ff8888ef6c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-312d71edcaca097c

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: 1,    6;
  2: 1,   20,     27;
  3: 1,   50,    201,      108;
  4: 1,  112,    978,     1504,      405;
  5: 1,  238,   3942,    12536,     9545,      1458;
  6: 1,  492,  14325,    81680,   126075,     54564,      5103;
  7: 1, 1002,  48879,   460420,  1242975,   1083834,    290353,    17496;
  8: 1, 2024, 160164,  2363056, 10243550,  15452424,   8354836,  1467296,   59049;
  9: 1, 4070, 510852, 11374192, 74848310, 178760364, 166650820, 59480288, 7134273, 196830;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+4*n+3)*x^2*P(n-2,x) - ((n+3)*x^2-(n+3)*x^3)*d/dx P(n-2,x) + (1+(2*n+4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 3)*x^2*P[n - 2, x] - ((n + 3)*x^2 - (n + 3)*x^3)*D[P[n - 2, x], x] + (1 + (2*n + 4)*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; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 3)*x^2*v[i - 2] - ((n + 3)*x^2 - (n + 3)*x^3)*deriv(v[i - 2], x) + (1 + (2*n + 4)*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 (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of 3-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[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.