A156365 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(-6 - 5\,x + 4\,x^{2} - 12\,x^{3} - 16\,x^{4} - 4\,n - 10\,n x - 22\,n x^{2} - 28\,n x^{3} - 16\,n x^{4} - 8\,n^{2} x - 16\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - x^{2} + 4\,x^{3} + 12\,x^{4} + 16\,x^{5} - 2\,n x - 4\,n x^{2} + 4\,n x^{3} + 24\,n x^{4} - 4\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(5 + 15\,x + 10\,x^{2} + 8\,x^{3} + 6\,n + 10\,n x + 20\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} - 8\,x^{3} + 2\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 9 + 16\,x + 4\,x^{2}\)

\(P_{1}(x) = 23 + 110\,x + 68\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	9	16	4								29
1	23	110	68	8							209
2	53	580	888	272	16						1809
3	115	2530	8456	6064	1072	32					18269
4	241	9768	62748	95552	38256	4224	64				210853
5	495	34838	392028	1124568	936784	232224	16704	128			2737769
6	1005	117868	2176048	10702032	16616480	8432960	1384704	66304	256		39497657
7	2027	384746	11115968	87504192	230622304	216725056	71996928	8198144	263936	512	626813813

All 15 rows

Technical details

Node id: n-5bccacf22ff4e1b7 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6093ad2f7ab921a1

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 2)
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:    9,      16,         4;
  1:   23,     110,        68,          8;
  2:   53,     580,       888,        272,          16;
  3:  115,    2530,      8456,       6064,        1072,          32;
  4:  241,    9768,     62748,      95552,       38256,        4224,          64;
  5:  495,   34838,    392028,    1124568,      936784,      232224,       16704,         128;
  6: 1005,  117868,   2176048,   10702032,    16616480,     8432960,     1384704,       66304,        256;
  7: 2027,  384746,  11115968,   87504192,   230622304,   216725056,    71996928,     8198144,     263936,       512;
  8: 4073, 1225168,  53494260,  640867200,  2671689504,  4242611712,  2597564544,   594647040,   48456960,   1052672,    1024;
  9: 8167, 3834334, 246417524, 4325883816, 27063061344, 67605110976, 69822139008, 29353143552, 4811920128, 286684672, 4203520, 2048;

%F Formula

The row polynomials satisfy (2*n+1+4*x)*P(n,x) = -(4*n+6+(8*n^2+10*n+5)*x+(8*n^3+16*n^2+22*n-4)*x^2+(24*n^2+28*n+12)*x^3+(16*n+16)*x^4)*P(n-2,x) - ((2*n+3)*x+(4*n^2+4*n+1)*x^2-(8*n^2+4*n+4)*x^3-(24*n+12)*x^4-16*x^5)*d/dx P(n-2,x) + (6*n+5+(8*n^2+10*n+15)*x+(20*n+10)*x^2+8*x^3)*P(n-1,x) + ((2*n+1)*x-(4*n-2)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 9 + 16*x + 4*x^2, P(1,x) = 23 + 110*x + 68*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 9 + 16*x + 4*x^2;
P[1, x_] := 23 + 110*x + 68*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 6 + (8*n^2 + 10*n + 5)*x + (8*n^3 + 16*n^2 + 22*n - 4)*x^2 + (24*n^2 + 28*n + 12)*x^3 + (16*n + 16)*x^4)*P[n - 2, x] - ((2*n + 3)*x + (4*n^2 + 4*n + 1)*x^2 - (8*n^2 + 4*n + 4)*x^3 - (24*n + 12)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (6*n + 5 + (8*n^2 + 10*n + 15)*x + (20*n + 10)*x^2 + 8*x^3)*P[n - 1, x] + ((2*n + 1)*x - (4*n - 2)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(2*n + 1 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 16*x + 4*x^2; v[2] = 23 + 110*x + 68*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 6 + (8*n^2 + 10*n + 5)*x + (8*n^3 + 16*n^2 + 22*n - 4)*x^2 + (24*n^2 + 28*n + 12)*x^3 + (16*n + 16)*x^4)*v[i - 2] - ((2*n + 3)*x + (4*n^2 + 4*n + 1)*x^2 - (8*n^2 + 4*n + 4)*x^3 - (24*n + 12)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (6*n + 5 + (8*n^2 + 10*n + 15)*x + (20*n + 10)*x^2 + 8*x^3)*v[i - 1] + ((2*n + 1)*x - (4*n - 2)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(2*n + 1 + 4*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 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 T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A142075 → D+1

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.