A166344 → θ+1

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 1 + 12\,x + 3\,x^{2}\)

\(P_{2}(x) = 1 + 30\,x + 45\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	1	12	3															16
2	1	30	45	4														80
3	1	68	270	136	5													480
4	1	146	1218	1624	365	6												3360
5	1	304	4749	12992	7915	912	7											26880
6	1	622	16983	83628	104535	33966	2177	8										241920
7	1	1260	57480	470376	1045350	705564	134120	5040	9									2419200
8	1	2538	187560	2417984	8754570	10505484	4231472	500160	11421	10								26611200
9	1	5096	596943	11672576	64930210	126065808	90902294	23345152	1790829	25480	11							319334400
10	1	10214	1867269	53853620	441376810	1301808612	1518776714	706202896	121170645	6224230	56177	12						4151347200
11	1	20452	5770338	240311400	2812737275	12048069048	21262873996	16064092064	5062927095	600778500	21157906	122712	13					58118860800
12	1	40930	17679150	1045877224	17067578875	102665976858	261288762420	298615728480	153998965287	34135157750	2876162366	70716600	266045	14				871782912000
13	1	81888	53835585	4466375296	99718706205	820806807648	2907014093949	4777851655680	3737589549363	1368011346080	219381153651	13399125888	233287535	573216	15			13948526592000
14	1	163806	163226403	18798651324	565589160465	6241127767794	29925178080203	68123529640344	76638970845387	42750254400290	11442067574289	1357413985116	61095616803	761723214	1228545	16		237124952064000
15	1	327644	493364844	78241498152	3133423886340	45591250517460	289570719581444	886990272325072	1379501475216966	1108737840406340	455039702199412	91182501034920	8146902104484	273845243532	2466824220	2621152	17	4268249137152000

Technical details

Node id: n-77ba82050b23ea81 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-8d82c7a5e5048ef9

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,    12,       3;
   2: 1,    30,      45,        4;
   3: 1,    68,     270,      136,         5;
   4: 1,   146,    1218,     1624,       365,          6;
   5: 1,   304,    4749,    12992,      7915,        912,          7;
   6: 1,   622,   16983,    83628,    104535,      33966,       2177,         8;
   7: 1,  1260,   57480,   470376,   1045350,     705564,     134120,      5040,         9;
   8: 1,  2538,  187560,  2417984,   8754570,   10505484,    4231472,    500160,     11421,      10;
   9: 1,  5096,  596943, 11672576,  64930210,  126065808,   90902294,  23345152,   1790829,   25480,    11;
  10: 1, 10214, 1867269, 53853620, 441376810, 1301808612, 1518776714, 706202896, 121170645, 6224230, 56177, 12;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+2*n)*x^2*P(n-2,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-2,x) + (1+(2*n+2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 12*x + 3*x^2, P(2,x) = 1 + 30*x + 45*x^2 + 4*x^3.

%t Mathematica

P[1, x_] := 1 + 12*x + 3*x^2;
P[2, x_] := 1 + 30*x + 45*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n)*x^2*P[n - 2, x] - ((n + 2)*x^2 - (n + 2)*x^3)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 12*x + 3*x^2; v[2] = 1 + 30*x + 45*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = -(n^2 + 2*n)*x^2*v[i - 2] - ((n + 2)*x^2 - (n + 2)*x^3)*deriv(v[i - 2], x) + (1 + (2*n + 2)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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 T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
    
    ↑ Liu–Wang theorem (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)
  - ↑ Parent-row provers of polytool (above)

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.