A136532 → D+1

Proved for all n all roots real interlacing (weak)

Recurrence

\(\left(2 + 3\,n + n^{2}\right)\,P_{n}(x) = \left(6\,n^{2} + 5\,n^{3} - 5\,n^{4} - 5\,n^{5} - n^{6}\right)\,P_{n-4}(x) + \left(-12\,n - 12\,n x + 2\,n^{2} - 22\,n^{2} x + 32\,n^{3} - 12\,n^{3} x + 22\,n^{4} - 2\,n^{4} x + 4\,n^{5}\right)\,P_{n-3}(x) + \left(-6 - 6\,x^{2} - 17\,n + 24\,n x - 5\,n x^{2} - 47\,n^{2} + 20\,n^{2} x - n^{2} x^{2} - 32\,n^{3} + 4\,n^{3} x - 6\,n^{4}\right)\,P_{n-2}(x) + \left(6 - 6\,x + 20\,n - 8\,n x + 18\,n^{2} - 2\,n^{2} x + 4\,n^{3}\right)\,P_{n-1}(x)\)

\(P_{1}(x) = -24 - 8\,x + 4\,x^{2}\)

\(P_{2}(x) = -125 + 35\,x + 35\,x^{2} - 5\,x^{3}\)

\(P_{3}(x) = -552 + 756\,x + 138\,x^{2} - 84\,x^{3} + 6\,x^{4}\)

\(P_{4}(x) = -1617 + 8435\,x - 1036\,x^{2} - 840\,x^{3} + 161\,x^{4} - 7\,x^{5}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	-24	-8	4															-28
2	-125	35	35	-5														-60
3	-552	756	138	-84	6													264
4	-1617	8435	-1036	-840	161	-7												5096
5	7040	80368	-34856	-4448	2656	-272	8											50496
6	228735	702567	-607383	46233	31905	-6471	423	-9										396000
7	3585240	5470300	-9012290	2151660	215750	-118300	13530	-620	10									2305280
8	48536587	30676679	-125269738	51557154	-2749736	-1687268	335874	-25498	869	-11								1374912
9	624578592	-88851960	-1677493164	1046342880	-166939656	-14055216	6924960	-812064	44520	-1176	12							-270262272
10	7831769595	-7744368853	-21640627489	19878278823	-5126396106	204786582	115172694	-21843354	1755663	-73281	1547	-13						-6501544192
11	95396903000	-200008430972	-261828402542	366618622412	-132541452330	15615565896	1149313620	-506530248	58656612	-3490060	115066	-1988	14					-116049131520
12	1095241536615	-4230884895525	-2718845522040	6663814015260	-3195236168925	587452419315	-18356881320	-9660663360	1724479245	-140553375	6494400	-173820	2505	-15				-1824885911040
13	10774931831808	-83329679830304	-15483433998032	119917838732736	-74655452695424	18498384356896	-1710552608208	-112943754624	44640085632	-5012138208	308767888	-11453504	254208	-3104	16			-26060982452224
14	55070381210263	-1591060647282865	333755019652229	2129894705412005	-1721706966426289	540908722873799	-76398426914795	1926348557493	962476174605	-160988997323	13007652119	-632860105	19317389	-361675	3791	-17		-326796981989376
15	-1382516849157480	-29881299667357332	18251142658791174	36892037742279372	-39557670715245834	15280861251656268	-2840459317641666	214790271202188	12947392191294	-4613786532828	497659929426	-30932360316	1225182114	-31371228	502506	-4572	18	-3014313097936896

Technical details

Node id: n-c6a8b7ddc1412b03 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-5dc0e20091c9a09f

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): no
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:        -24,          -8,            4;
   2:       -125,          35,           35,          -5;
   3:       -552,         756,          138,         -84,           6;
   4:      -1617,        8435,        -1036,        -840,         161,        -7;
   5:       7040,       80368,       -34856,       -4448,        2656,      -272,         8;
   6:     228735,      702567,      -607383,       46233,       31905,     -6471,       423,        -9;
   7:    3585240,     5470300,     -9012290,     2151660,      215750,   -118300,     13530,      -620,      10;
   8:   48536587,    30676679,   -125269738,    51557154,    -2749736,  -1687268,    335874,    -25498,     869,    -11;
   9:  624578592,   -88851960,  -1677493164,  1046342880,  -166939656, -14055216,   6924960,   -812064,   44520,  -1176,   12;
  10: 7831769595, -7744368853, -21640627489, 19878278823, -5126396106, 204786582, 115172694, -21843354, 1755663, -73281, 1547, -13;

%F Formula

The row polynomials satisfy (n^2+3*n+2)*P(n,x) = -(n^6+5*n^5+5*n^4-5*n^3-6*n^2)*P(n-4,x) + (4*n^5+22*n^4+32*n^3+2*n^2-12*n-(2*n^4+12*n^3+22*n^2+12*n)*x)*P(n-3,x) - (6*n^4+32*n^3+47*n^2+17*n+6-(4*n^3+20*n^2+24*n)*x+(n^2+5*n+6)*x^2)*P(n-2,x) + (4*n^3+18*n^2+20*n+6-(2*n^2+8*n+6)*x)*P(n-1,x) for n >= 5, with P(1,x) = -24 - 8*x + 4*x^2, P(2,x) = -125 + 35*x + 35*x^2 - 5*x^3, P(3,x) = -552 + 756*x + 138*x^2 - 84*x^3 + 6*x^4, P(4,x) = -1617 + 8435*x - 1036*x^2 - 840*x^3 + 161*x^4 - 7*x^5.

%t Mathematica

P[1, x_] := -24 - 8*x + 4*x^2;
P[2, x_] := -125 + 35*x + 35*x^2 - 5*x^3;
P[3, x_] := -552 + 756*x + 138*x^2 - 84*x^3 + 6*x^4;
P[4, x_] := -1617 + 8435*x - 1036*x^2 - 840*x^3 + 161*x^4 - 7*x^5;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 5*n^5 + 5*n^4 - 5*n^3 - 6*n^2)*P[n - 4, x] + (4*n^5 + 22*n^4 + 32*n^3 + 2*n^2 - 12*n - (2*n^4 + 12*n^3 + 22*n^2 + 12*n)*x)*P[n - 3, x] - (6*n^4 + 32*n^3 + 47*n^2 + 17*n + 6 - (4*n^3 + 20*n^2 + 24*n)*x + (n^2 + 5*n + 6)*x^2)*P[n - 2, x] + (4*n^3 + 18*n^2 + 20*n + 6 - (2*n^2 + 8*n + 6)*x)*P[n - 1, x])/(n^2 + 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = -24 - 8*x + 4*x^2; v[2] = -125 + 35*x + 35*x^2 - 5*x^3; v[3] = -552 + 756*x + 138*x^2 - 84*x^3 + 6*x^4; v[4] = -1617 + 8435*x - 1036*x^2 - 840*x^3 + 161*x^4 - 7*x^5; for(n = 5, N, my(i = n); v[i] = (-(n^6 + 5*n^5 + 5*n^4 - 5*n^3 - 6*n^2)*v[i - 4] + (4*n^5 + 22*n^4 + 32*n^3 + 2*n^2 - 12*n - (2*n^4 + 12*n^3 + 22*n^2 + 12*n)*x)*v[i - 3] - (6*n^4 + 32*n^3 + 47*n^2 + 17*n + 6 - (4*n^3 + 20*n^2 + 24*n)*x + (n^2 + 5*n + 6)*x^2)*v[i - 2] + (4*n^3 + 18*n^2 + 20*n + 6 - (2*n^2 + 8*n + 6)*x)*v[i - 1])/(n^2 + 3*n + 2)); 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 (−∞, +∞)
by 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, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023