A142460 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-14 + 10\,x + 5\,n\right)\,P_{n}(x) = \left(54 - 357\,x + 1914\,x^{2} - 2165\,x^{3} + 450\,x^{4} - 30\,n + 255\,n x - 2240\,n x^{2} + 1850\,n x^{3} - 250\,n x^{4} - 50\,n^{2} x + 925\,n^{2} x^{2} - 375\,n^{2} x^{3} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(45\,x - 725\,x^{2} + 1605\,x^{3} - 1175\,x^{4} + 250\,x^{5} - 25\,n x + 600\,n x^{2} - 950\,n x^{3} + 375\,n x^{4} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-93 + 392\,x - 325\,x^{2} + 50\,x^{3} + 35\,n - 255\,n x + 125\,n x^{2} + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-70\,x + 120\,x^{2} - 50\,x^{3} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	13	14	1						28
3	84	249	86	1					420
4	515	4166	3368	518	1				8568
5	3106	59725	113240	40730	3110	1			219912
6	18657	771886	3094375	2412380	469895	18662	1		6785856
7	111968	9367609	71838726	112018175	45542740	5299623	111974	1	244290816

All 15 rows

Technical details

Node id: n-45c51bfdab584967 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-46d369865b1f7869

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:       2,          1;
  2:      13,         14,           1;
  3:      84,        249,          86,            1;
  4:     515,       4166,        3368,          518,            1;
  5:    3106,      59725,      113240,        40730,         3110,           1;
  6:   18657,     771886,     3094375,      2412380,       469895,       18662,           1;
  7:  111968,    9367609,    71838726,    112018175,     45542740,     5299623,      111974,         1;
  8:  671839,  109425854,  1490721526,   4242592186,   3334860200,   805043554,    59079874,    671846,       1;
  9: 4031070, 1248697581, 28660907288, 138179492248, 193249951356, 88285473290, 13702524136, 654581808, 4031078, 1;

%F Formula

The row polynomials satisfy (5*n-14+10*x)*P(n,x) = -(30*n-54+(50*n^2-255*n+357)*x+(125*n^3-925*n^2+2240*n-1914)*x^2+(375*n^2-1850*n+2165)*x^3+(250*n-450)*x^4)*P(n-2,x) - ((25*n-45)*x+(125*n^2-600*n+725)*x^2-(125*n^2-950*n+1605)*x^3-(375*n-1175)*x^4-250*x^5)*d/dx P(n-2,x) + (35*n-93+(50*n^2-255*n+392)*x+(125*n-325)*x^2+50*x^3)*P(n-1,x) + ((25*n-70)*x-(25*n-120)*x^2-50*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 54 + (50*n^2 - 255*n + 357)*x + (125*n^3 - 925*n^2 + 2240*n - 1914)*x^2 + (375*n^2 - 1850*n + 2165)*x^3 + (250*n - 450)*x^4)*P[n - 2, x] - ((25*n - 45)*x + (125*n^2 - 600*n + 725)*x^2 - (125*n^2 - 950*n + 1605)*x^3 - (375*n - 1175)*x^4 - 250*x^5)*D[P[n - 2, x], x] + (35*n - 93 + (50*n^2 - 255*n + 392)*x + (125*n - 325)*x^2 + 50*x^3)*P[n - 1, x] + ((25*n - 70)*x - (25*n - 120)*x^2 - 50*x^3)*D[P[n - 1, x], x])/(5*n - 14 + 10*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n - 54 + (50*n^2 - 255*n + 357)*x + (125*n^3 - 925*n^2 + 2240*n - 1914)*x^2 + (375*n^2 - 1850*n + 2165)*x^3 + (250*n - 450)*x^4)*v[i - 2] - ((25*n - 45)*x + (125*n^2 - 600*n + 725)*x^2 - (125*n^2 - 950*n + 1605)*x^3 - (375*n - 1175)*x^4 - 250*x^5)*deriv(v[i - 2], x) + (35*n - 93 + (50*n^2 - 255*n + 392)*x + (125*n - 325)*x^2 + 50*x^3)*v[i - 1] + ((25*n - 70)*x - (25*n - 120)*x^2 - 50*x^3)*deriv(v[i - 1], x))/(5*n - 14 + 10*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 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*…?
  - 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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*…?
    
    ↑ 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.