A144699 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + n\right)\,P_{n}(x) = \left(-6 - 13\,x - 18\,x^{2} - 17\,x^{3} - 6\,x^{4} - 2\,n - 9\,n x - 20\,n x^{2} - 14\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x - 7\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - 5\,x^{2} + x^{3} + 5\,x^{4} + 2\,x^{5} - n x - 4\,n x^{2} + 2\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(7 + 16\,x + 11\,x^{2} + 2\,x^{3} + 3\,n + 9\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 6 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	5														11
2	17	66	25													108
3	40	381	546	125												1092
4	87	1658	5364	4026	625											11760
5	182	6273	37864	60730	27906	3125										136080
6	373	21906	223143	641380	604827	186186	15625									1693440
7	756	72797	1176234	5498355	9045296	5537091	1210866	78125								22619520
8	1523	234258	5757850	41154470	107595684	112675150	47782574	7733946	390625							323326080
9	3058	737793	26773440	280391664	1096392780	1796198082	1283661792	394865640	48747426	1953125						4929724800
10	6129	2289818	119991381	1784853768	10002152034	24154411596	26700160242	13677573864	3158025837	304203306	9765625					79913433600
11	12272	7034925	523441426	10801088889	84049013304	286757227722	462559145580	363399606954	138402644616	24626229001	1883813586	48828125				1373058086400
12	24559	21460650	2237914176	62893414706	663306848379	3097280822292	6990042076584	7955827824564	4616652946149	1344632115266	188255844984	11595850266	244140625			24932991283200
13	49134	65143265	9423643656	355459773270	4984650928150	31081277698227	95066191149120	150348747809028	125724807707130	55510126643895	12644414843704	1416539991750	71039945346	1220703125		477213966028800
14	98285	197051202	39225121987	1962511185676	36032543335365	294209252888390	1189671109255731	2532525537135816	2928269875069023	1855799555086510	638289054446625	115797886966892	10524143544047	433563890826	6103515625	9603560558592000

Technical details

Node id: n-778c27541fff7d28 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c0dc793f2abea0c9

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:    6,      5;
  2:   17,     66,       25;
  3:   40,    381,      546,       125;
  4:   87,   1658,     5364,      4026,        625;
  5:  182,   6273,    37864,     60730,      27906,       3125;
  6:  373,  21906,   223143,    641380,     604827,     186186,      15625;
  7:  756,  72797,  1176234,   5498355,    9045296,    5537091,    1210866,     78125;
  8: 1523, 234258,  5757850,  41154470,  107595684,  112675150,   47782574,   7733946,   390625;
  9: 3058, 737793, 26773440, 280391664, 1096392780, 1796198082, 1283661792, 394865640, 48747426, 1953125;

%F Formula

The row polynomials satisfy (n+2+2*x)*P(n,x) = -(2*n+6+(2*n^2+9*n+13)*x+(n^3+7*n^2+20*n+18)*x^2+(3*n^2+14*n+17)*x^3+(2*n+6)*x^4)*P(n-2,x) - ((n+3)*x+(n^2+4*n+5)*x^2-(n^2+2*n+1)*x^3-(3*n+5)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n+7+(2*n^2+9*n+16)*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) = 6 + 5*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 6 + (2*n^2 + 9*n + 13)*x + (n^3 + 7*n^2 + 20*n + 18)*x^2 + (3*n^2 + 14*n + 17)*x^3 + (2*n + 6)*x^4)*P[n - 2, x] - ((n + 3)*x + (n^2 + 4*n + 5)*x^2 - (n^2 + 2*n + 1)*x^3 - (3*n + 5)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n + 7 + (2*n^2 + 9*n + 16)*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] = 6 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 6 + (2*n^2 + 9*n + 13)*x + (n^3 + 7*n^2 + 20*n + 18)*x^2 + (3*n^2 + 14*n + 17)*x^3 + (2*n + 6)*x^4)*v[i - 2] - ((n + 3)*x + (n^2 + 4*n + 5)*x^2 - (n^2 + 2*n + 1)*x^3 - (3*n + 5)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n + 7 + (2*n^2 + 9*n + 16)*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 of 5-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.