A257607 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + n\right)\,P_{n}(x) = \left(-90 - 105\,x - 54\,x^{2} - 25\,x^{3} - 6\,x^{4} - 30\,n - 45\,n x - 40\,n x^{2} - 14\,n x^{3} - 2\,n x^{4} - 10\,n^{2} x - 7\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-15\,x - x^{2} + 9\,x^{3} + 5\,x^{4} + 2\,x^{5} - 5\,n x + 2\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(27 + 32\,x + 11\,x^{2} + 2\,x^{3} + 11\,n + 9\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 10 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	10	5							15
2	85	110	25						220
3	660	1605	910	125					3300
4	4835	19190	20120	6710	625				51480
5	34010	203105	344440	209650	46510	3125			840840
6	232185	1978030	5014015	4900700	1963535	310310	15625		14414400
7	1549360	18132725	65176830	94743475	60602420	17158155	2018110	78125	259459200

All 15 rows

Technical details

Node id: n-55cb0598f1d8c8ee · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-59069e47282240be

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:       10,          5;
  2:       85,        110,         25;
  3:      660,       1605,        910,         125;
  4:     4835,      19190,      20120,        6710,         625;
  5:    34010,     203105,     344440,      209650,       46510,        3125;
  6:   232185,    1978030,    5014015,     4900700,     1963535,      310310,      15625;
  7:  1549360,   18132725,   65176830,    94743475,    60602420,    17158155,    2018110,      78125;
  8: 10155535,  158762510,  779189830,  1599203050,  1515561560,   681889810,  142853170,   12889910,   390625;
  9: 65620710, 1341008625, 8734659160, 24373297640, 32511952140, 21610635970, 7169356520, 1148011200, 81245710, 1953125;

%F Formula

The row polynomials satisfy (n+2+2*x)*P(n,x) = -(30*n+90+(10*n^2+45*n+105)*x+(n^3+7*n^2+40*n+54)*x^2+(3*n^2+14*n+25)*x^3+(2*n+6)*x^4)*P(n-2,x) - ((5*n+15)*x+(n^2+1)*x^2-(n^2+2*n+9)*x^3-(3*n+5)*x^4-2*x^5)*d/dx P(n-2,x) + (11*n+27+(2*n^2+9*n+32)*x+(5*n+11)*x^2+2*x^3)*P(n-1,x) + ((n+2)*x-n*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 5*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 10 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 90 + (10*n^2 + 45*n + 105)*x + (n^3 + 7*n^2 + 40*n + 54)*x^2 + (3*n^2 + 14*n + 25)*x^3 + (2*n + 6)*x^4)*P[n - 2, x] - ((5*n + 15)*x + (n^2 + 1)*x^2 - (n^2 + 2*n + 9)*x^3 - (3*n + 5)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (11*n + 27 + (2*n^2 + 9*n + 32)*x + (5*n + 11)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 2)*x - n*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 2 + 2*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] = 10 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n + 90 + (10*n^2 + 45*n + 105)*x + (n^3 + 7*n^2 + 40*n + 54)*x^2 + (3*n^2 + 14*n + 25)*x^3 + (2*n + 6)*x^4)*v[i - 2] - ((5*n + 15)*x + (n^2 + 1)*x^2 - (n^2 + 2*n + 9)*x^3 - (3*n + 5)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (11*n + 27 + (2*n^2 + 9*n + 32)*x + (5*n + 11)*x^2 + 2*x^3)*v[i - 1] + ((n + 2)*x - n*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 2 + 2*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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.?
  - 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.?
    
    ↑ 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.