A060187 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-1 + 10\,x + 15\,x^{2} + 2\,n - 24\,n x - 46\,n x^{2} + 8\,n^{2} x + 36\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 4\,x^{2} + 6\,x^{3} + 4\,n x + 12\,n x^{2} - 16\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 18\,x - 24\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	2														3
3	1	12	3													16
4	1	46	69	4												120
5	1	152	690	304	5											1152
6	1	474	5046	6728	1185	6										13440
7	1	1444	31629	94192	52715	4332	7									184320
8	1	4358	181971	1038892	1298615	363942	15253	8								2903040
9	1	13104	994836	9941152	23375070	14911728	2321284	52416	9							51609600
10	1	39346	5269020	86831888	347066470	416479764	151955804	14050720	177057	10						1021870080
11	1	118076	27348423	713203616	4534616410	9162554808	6348462974	1426407232	82045269	590380	11					22295347200
12	1	354270	140107281	5612322900	54184115490	171528118884	200116138698	86694584784	12627726525	467024270	1948485	12				531372441600
13	1	1062856	711695910	42835644752	606918901035	2862123386544	5203019606148	3816164515392	1092454021863	107089111880	2609551670	6377136	13			13733933875200
14	1	3188618	3595148082	319776998744	6479616672175	43864502833098	117726414144444	134544473307936	65796754249647	12959233344350	879386746546	14380592328	20726017	14		382588157952000
15	1	9565908	18095313585	2348690199056	66710696266605	630244791289668	2400422690067549	4036334199238080	3086257744372563	1050407985482780	146763531786531	7046070597168	78413025535	66961356	15	11426632984166400

Technical details

Node id: n-88475cccc0c2813e · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2e37e4aadfbd49d5

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 -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;
   2: 1,     2;
   3: 1,    12,       3;
   4: 1,    46,      69,        4;
   5: 1,   152,     690,      304,         5;
   6: 1,   474,    5046,     6728,      1185,         6;
   7: 1,  1444,   31629,    94192,     52715,      4332,         7;
   8: 1,  4358,  181971,  1038892,   1298615,    363942,     15253,        8;
   9: 1, 13104,  994836,  9941152,  23375070,  14911728,   2321284,    52416,      9;
  10: 1, 39346, 5269020, 86831888, 347066470, 416479764, 151955804, 14050720, 177057, 10;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = (2*n-1+(8*n^2-24*n+10)*x-(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) + ((4*n-2)*x-(8*n^2-12*n+4)*x^2+(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) - (2-(8*n^2-24*n+18)*x)*P(n-1,x) + ((4*n-6)*x-(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (8*n^2 - 24*n + 10)*x - (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] + ((4*n - 2)*x - (8*n^2 - 12*n + 4)*x^2 + (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] - (2 - (8*n^2 - 24*n + 18)*x)*P[n - 1, x] + ((4*n - 6)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = ((2*n - 1 + (8*n^2 - 24*n + 10)*x - (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] + ((4*n - 2)*x - (8*n^2 - 12*n + 4)*x^2 + (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) - (2 - (8*n^2 - 24*n + 18)*x)*v[i - 1] + ((4*n - 6)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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 read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.