A375853 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 9\,x - 2\,x^{2} + 12\,n - 13\,n x + 2\,n x^{2} - 7\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(8 - 22\,x + 18\,x^{2} - 4\,x^{3} + 2\,n + 15\,n x - 27\,n x^{2} + 6\,n x^{3} + 3\,n^{2} - 11\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{3} + 6\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x - 16\,x^{2} + 14\,x^{3} - 4\,x^{4} - 4\,n x + 12\,n x^{2} - 8\,n x^{3} + 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	2															2
2	16	8														24
3	76	96	20													192
4	256	648	336	40												1280
5	686	2992	3036	896	70											7680
6	1568	10608	18304	10400	2016	112										43008
7	3192	31104	82680	79040	29160	4032	168									229376
8	5952	79152	301920	442000	272000	70992	7392	240								1179648
9	10362	180576	937992	1963840	1841100	795872	155496	12672	330							5898240
10	17072	377784	2568192	7307552	9767520	6403152	2059904	313632	20592	440						28835840
11	26884	736736	6354348	23632960	42850472	39938304	19406296	4841408	592020	32032	572					138412032
12	40768	1355640	14471600	68231800	161515200	203851760	140469280	52743600	10524800	1058200	48048	728				654311424
13	59878	2375568	30759300	179400000	537941250	885396960	824828760	438177600	131186250	21450000	1806948	69888	910			3053453312
14	85568	3993184	61671168	436113600	1616808000	3368403360	4087802880	2931384960	1239019200	303108000	41403648	2967744	99008	1120		14092861440
15	119408	6475776	117617968	991805568	4456522800	11473253760	17611841520	16435733760	9364199760	3227769600	658024848	76292736	4713488	137088	1360	64424509440

Technical details

Node id: n-63ccdddb32d321a1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ea07d872e5cb1eb5

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:     2;
   2:    16,      8;
   3:    76,     96,      20;
   4:   256,    648,     336,      40;
   5:   686,   2992,    3036,     896,      70;
   6:  1568,  10608,   18304,   10400,    2016,     112;
   7:  3192,  31104,   82680,   79040,   29160,    4032,     168;
   8:  5952,  79152,  301920,  442000,  272000,   70992,    7392,    240;
   9: 10362, 180576,  937992, 1963840, 1841100,  795872,  155496,  12672,   330;
  10: 17072, 377784, 2568192, 7307552, 9767520, 6403152, 2059904, 313632, 20592, 440;

%F Formula

The row polynomials satisfy (2*n^3-7*n^2+12*n-7+(4*n^2-13*n+9)*x+(2*n-2)*x^2)*P(n,x) = (2*n^3+3*n^2+2*n+8+(6*n^3-11*n^2+15*n-22)*x+(12*n^2-27*n+18)*x^2+(6*n-4)*x^3)*P(n-1,x) + ((4*n^2-4*n+6)*x-(4*n^2-12*n+16)*x^2-(8*n-14)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

%t Mathematica

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 3*n^2 + 2*n + 8 + (6*n^3 - 11*n^2 + 15*n - 22)*x + (12*n^2 - 27*n + 18)*x^2 + (6*n - 4)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n + 6)*x - (4*n^2 - 12*n + 16)*x^2 - (8*n - 14)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 7*n^2 + 12*n - 7 + (4*n^2 - 13*n + 9)*x + (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n^3 + 3*n^2 + 2*n + 8 + (6*n^3 - 11*n^2 + 15*n - 22)*x + (12*n^2 - 27*n + 18)*x^2 + (6*n - 4)*x^3)*v[i - 1] + ((4*n^2 - 4*n + 6)*x - (4*n^2 - 12*n + 16)*x^2 - (8*n - 14)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 7*n^2 + 12*n - 7 + (4*n^2 - 13*n + 9)*x + (2*n - 2)*x^2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
  - 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: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
    
    ↑ Liu–Wang theorem (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.