A046739 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\right)\,P_{n}(x) = \left(-16\,n - 20\,n x - 14\,n x^{2} + 9\,n x^{3} + 4\,n^{2} - 10\,n^{2} x - 14\,n^{2} x^{2} + 2\,n^{2} x^{3} + 12\,n^{3} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-4}(x) + \left(-8\,n x - 6\,n x^{2} + 9\,n x^{3} + 2\,n^{2} x - 6\,n^{2} x^{2} + 2\,n^{2} x^{3} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-4}'(x) + \left(24 + 6\,x + 35\,x^{2} - 25\,x^{3} - 22\,n - 16\,n x + 6\,n x^{2} - 9\,n x^{3} - 14\,n^{2} - 16\,n^{2} x - 25\,n^{2} x^{2} + 5\,n^{2} x^{3} + 12\,n^{3} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-24 + 14\,x + 41\,x^{2} - 12\,x^{3} - 9\,x^{4} + 6\,n - 34\,n x + 18\,n x^{2} + 11\,n x^{3} - 2\,n x^{4} + 18\,n^{2} - 22\,n^{2} x - 15\,n^{2} x^{2} + 8\,n^{2} x^{3} + 3\,n^{2} x^{4} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(-8\,x + 2\,x^{2} + 15\,x^{3} - 9\,x^{4} + 2\,n x - 8\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} + 6\,n^{2} x - 6\,n^{2} x^{2} - 3\,n^{2} x^{3} + 3\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(-12 + 84\,x - 25\,x^{2} - 14\,n + 36\,n x + 10\,n x^{2} - 16\,n x^{3} - 6\,n^{2} - 12\,n^{2} x - 5\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(8 + 20\,x - 35\,x^{2} - 4\,x^{3} + 16\,x^{4} - 2\,n + 4\,n x^{2} - 6\,n^{2} + 3\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(8\,x^{2} - 16\,x^{3} + 8\,x^{4}\right)\,P_{n-2}''(x) + \left(26 - 28\,x + 16\,x^{2} + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 8 + 9\,x + x^{2}\)

\(P_{2}(x) = 22 + 63\,x + 24\,x^{2} + x^{3}\)

\(P_{3}(x) = 52 + 373\,x + 314\,x^{2} + 55\,x^{3} + x^{4}\)

\(P_{4}(x) = 114 + 1739\,x + 3252\,x^{2} + 1265\,x^{3} + 118\,x^{4} + x^{5}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	8	9	1															18
2	22	63	24	1														110
3	52	373	314	55	1													795
4	114	1739	3252	1265	118	1												6489
5	240	6961	26254	21075	4556	245	1											59332
6	494	25335	173920	269165	115938	15379	500	1										600732
7	1004	86733	1003186	2748143	2207544	577297	49886	1011	1									6674805
8	2026	285123	5255628	23680761	32964102	15722175	2695536	157749	2034	1								80765135
9	4072	911881	25721942	180119131	405950628	330149981	101882866	12053919	490544	4081	1							1057289046
10	8166	2861615	119850072	1249307701	4312602378	5614201691	2919237004	618253905	52314974	1508375	8176	1						14890154058
11	16356	8862181	538620346	8086291543	40926320688	80823486777	66701242278	23606304315	3579648732	222294941	4602818	16367	1					224497707343
12	32738	27191355	2356247620	49655219681	356053789486	1021285836007	1270025059496	708889463517	178734284434	20027476931	930261836	13972153	32750	1				3607998868005
13	65504	82883169	10099968286	292767588259	2894573285852	11646582474149	20905462518938	17498818942119	6927018376024	1287980758185	109235068438	3849759147	42260372	65517	1			61576514013960
14	131038	251465703	42626656464	1672154547133	22304753712754	122421292439011	306284194572676	368498984220441	218009193191046	63454387162463	8937517129592	584500844213	15800671706	127501851	131052	1		1112225784377144
15	262108	760424317	177773916098	9313746857983	164675071227688	1205299640957729	4085320688576974	6821954779577443	5784517772899316	2510405042817925	552666344380634	60236130205511	3082360513216	64452413801	384012710	262123	1	21197714949305577

Technical details

Node id: n-614aff0891e33f9b · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e498d6ad1a16c710

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 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:    8,       9,         1;
   2:   22,      63,        24,          1;
   3:   52,     373,       314,         55,          1;
   4:  114,    1739,      3252,       1265,        118,          1;
   5:  240,    6961,     26254,      21075,       4556,        245,          1;
   6:  494,   25335,    173920,     269165,     115938,      15379,        500,         1;
   7: 1004,   86733,   1003186,    2748143,    2207544,     577297,      49886,      1011,        1;
   8: 2026,  285123,   5255628,   23680761,   32964102,   15722175,    2695536,    157749,     2034,       1;
   9: 4072,  911881,  25721942,  180119131,  405950628,  330149981,  101882866,  12053919,   490544,    4081,    1;
  10: 8166, 2861615, 119850072, 1249307701, 4312602378, 5614201691, 2919237004, 618253905, 52314974, 1508375, 8176, 1;

%F Formula

The row polynomials satisfy 8*P(n,x) = (12*n^3+4*n^2-16*n+(6*n^3-10*n^2-20*n)*x-(14*n^2+14*n)*x^2-(3*n^3-2*n^2-9*n)*x^3)*P(n-4,x) + ((6*n^3+2*n^2-8*n)*x-(6*n^2+6*n)*x^2-(3*n^3-2*n^2-9*n)*x^3)*d/dx P(n-4,x) + (12*n^3-14*n^2-22*n+24+(6*n^3-16*n^2-16*n+6)*x-(25*n^2-6*n-35)*x^2-(3*n^3-5*n^2+9*n+25)*x^3)*P(n-3,x) + (18*n^2+6*n-24+(6*n^3-22*n^2-34*n+14)*x-(15*n^2-18*n-41)*x^2-(3*n^3-8*n^2-11*n+12)*x^3+(3*n^2-2*n-9)*x^4)*d/dx P(n-3,x) + ((6*n^2+2*n-8)*x-(6*n^2+8*n-2)*x^2-(3*n^2-8*n-15)*x^3+(3*n^2-2*n-9)*x^4)*d^2/dx^2 P(n-3,x) - (6*n^2+14*n+12+(12*n^2-36*n-84)*x+(5*n^2-10*n+25)*x^2+16*n*x^3)*P(n-2,x) - (6*n^2+2*n-8-20*x-(3*n^2+4*n-35)*x^2+4*x^3-16*x^4)*d/dx P(n-2,x) + (8*x^2-16*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (6*n+26+(16*n-28)*x+16*x^2)*P(n-1,x) for n >= 5, with P(1,x) = 8 + 9*x + x^2, P(2,x) = 22 + 63*x + 24*x^2 + x^3, P(3,x) = 52 + 373*x + 314*x^2 + 55*x^3 + x^4, P(4,x) = 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5.

%t Mathematica

P[1, x_] := 8 + 9*x + x^2;
P[2, x_] := 22 + 63*x + 24*x^2 + x^3;
P[3, x_] := 52 + 373*x + 314*x^2 + 55*x^3 + x^4;
P[4, x_] := 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^3 + 4*n^2 - 16*n + (6*n^3 - 10*n^2 - 20*n)*x - (14*n^2 + 14*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*P[n - 4, x] + ((6*n^3 + 2*n^2 - 8*n)*x - (6*n^2 + 6*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*D[P[n - 4, x], x] + (12*n^3 - 14*n^2 - 22*n + 24 + (6*n^3 - 16*n^2 - 16*n + 6)*x - (25*n^2 - 6*n - 35)*x^2 - (3*n^3 - 5*n^2 + 9*n + 25)*x^3)*P[n - 3, x] + (18*n^2 + 6*n - 24 + (6*n^3 - 22*n^2 - 34*n + 14)*x - (15*n^2 - 18*n - 41)*x^2 - (3*n^3 - 8*n^2 - 11*n + 12)*x^3 + (3*n^2 - 2*n - 9)*x^4)*D[P[n - 3, x], x] + ((6*n^2 + 2*n - 8)*x - (6*n^2 + 8*n - 2)*x^2 - (3*n^2 - 8*n - 15)*x^3 + (3*n^2 - 2*n - 9)*x^4)*D[P[n - 3, x], {x, 2}] - (6*n^2 + 14*n + 12 + (12*n^2 - 36*n - 84)*x + (5*n^2 - 10*n + 25)*x^2 + 16*n*x^3)*P[n - 2, x] - (6*n^2 + 2*n - 8 - 20*x - (3*n^2 + 4*n - 35)*x^2 + 4*x^3 - 16*x^4)*D[P[n - 2, x], x] + (8*x^2 - 16*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (6*n + 26 + (16*n - 28)*x + 16*x^2)*P[n - 1, x])/(8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 8 + 9*x + x^2; v[2] = 22 + 63*x + 24*x^2 + x^3; v[3] = 52 + 373*x + 314*x^2 + 55*x^3 + x^4; v[4] = 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5; for(n = 5, N, my(i = n); v[i] = ((12*n^3 + 4*n^2 - 16*n + (6*n^3 - 10*n^2 - 20*n)*x - (14*n^2 + 14*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*v[i - 4] + ((6*n^3 + 2*n^2 - 8*n)*x - (6*n^2 + 6*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*deriv(v[i - 4], x) + (12*n^3 - 14*n^2 - 22*n + 24 + (6*n^3 - 16*n^2 - 16*n + 6)*x - (25*n^2 - 6*n - 35)*x^2 - (3*n^3 - 5*n^2 + 9*n + 25)*x^3)*v[i - 3] + (18*n^2 + 6*n - 24 + (6*n^3 - 22*n^2 - 34*n + 14)*x - (15*n^2 - 18*n - 41)*x^2 - (3*n^3 - 8*n^2 - 11*n + 12)*x^3 + (3*n^2 - 2*n - 9)*x^4)*deriv(v[i - 3], x) + ((6*n^2 + 2*n - 8)*x - (6*n^2 + 8*n - 2)*x^2 - (3*n^2 - 8*n - 15)*x^3 + (3*n^2 - 2*n - 9)*x^4)*deriv(deriv(v[i - 3], x), x) - (6*n^2 + 14*n + 12 + (12*n^2 - 36*n - 84)*x + (5*n^2 - 10*n + 25)*x^2 + 16*n*x^3)*v[i - 2] - (6*n^2 + 2*n - 8 - 20*x - (3*n^2 + 4*n - 35)*x^2 + 4*x^3 - 16*x^4)*deriv(v[i - 2], x) + (8*x^2 - 16*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n + 26 + (16*n - 28)*x + 16*x^2)*v[i - 1])/(8)); 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, related to number of permutations of [n] with 0 successions and k rises.?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.