A142462 → D

nonnegative coefficients

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

Recurrence

\(\left(-13 + 7\,n\right)\,P_{n}(x) = \left(48 - 240\,x + 1560\,x^{2} - 56\,n + 364\,n x - 3206\,n x^{2} - 98\,n^{2} x + 1911\,n^{2} x^{2} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(42\,x - 588\,x^{2} + 546\,x^{3} - 49\,n x + 980\,n x^{2} - 931\,n x^{3} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-110 + 338\,x + 63\,n - 364\,n x + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-91\,x + 91\,x^{2} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	16	2													18
3	143	286	3												432
4	1166	8580	3498	4											13248
5	9357	180004	270006	37428	5										496800
6	74892	3242766	11880264	6485532	374460	6									22057920
7	599179	54033714	402426129	536568172	135084285	3595074	7								1132306560
8	4793482	862035104	11771195394	31120953976	19618658990	2586105312	33554374	8							65997296640
9	38347913	13409874760	314567562876	1467525044216	1834406305270	629135125752	46934561660	306783304	9						4306323605760
10	306783368	205519783482	7926226990272	60593433220488	132077253979440	90890149830732	18494529643968	822079133928	2761050312	10					311012260416000
11	2454267015	3122065023334	191948935273695	2285624696078952	8011678178861070	9614013814633284	3999843218138166	511863827396520	14049292605003	24542670150	11				24632171024947200
12	19634136198	47179481266140	4522594818261354	80871235267090428	431271957943933020	826805188058462424	603780741121506228	161742470534180856	13567784454784062	235897406330700	215975498178	12			2122845284695449600
13	157073089669	710755144238988	104521700756593698	2731193913903924620	21287865998761781715	61428940563149865432	71667097323674843004	34060585598018850744	6145186306283830395	348405669188645660	3909153293314434	1884877076028	13		197778419024126054400
14	1256584717444	10688029588828550	2382635768521022952	89099344508171350724	984858689067738111340	4097534478530750082882	7166709732367484300400	5463379304707666777176	1772745640321928600412	222748361270428376810	8736331151243750824	64128177532971300	16335601326772	14	19808269351493240217600

Technical details

Node id: n-c213a6aaa87bf34f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d58ee5ca4c84ca3d

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        0;
  1:        1;
  2:       16,           2;
  3:      143,         286,            3;
  4:     1166,        8580,         3498,             4;
  5:     9357,      180004,       270006,         37428,             5;
  6:    74892,     3242766,     11880264,       6485532,        374460,            6;
  7:   599179,    54033714,    402426129,     536568172,     135084285,      3595074,           7;
  8:  4793482,   862035104,  11771195394,   31120953976,   19618658990,   2586105312,    33554374,         8;
  9: 38347913, 13409874760, 314567562876, 1467525044216, 1834406305270, 629135125752, 46934561660, 306783304, 9;

%F Formula

The row polynomials satisfy (7*n-13)*P(n,x) = -(56*n-48+(98*n^2-364*n+240)*x+(343*n^3-1911*n^2+3206*n-1560)*x^2)*P(n-2,x) - ((49*n-42)*x+(343*n^2-980*n+588)*x^2-(343*n^2-931*n+546)*x^3)*d/dx P(n-2,x) + (63*n-110+(98*n^2-364*n+338)*x)*P(n-1,x) + ((49*n-91)*x-(49*n-91)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(56*n - 48 + (98*n^2 - 364*n + 240)*x + (343*n^3 - 1911*n^2 + 3206*n - 1560)*x^2)*P[n - 2, x] - ((49*n - 42)*x + (343*n^2 - 980*n + 588)*x^2 - (343*n^2 - 931*n + 546)*x^3)*D[P[n - 2, x], x] + (63*n - 110 + (98*n^2 - 364*n + 338)*x)*P[n - 1, x] + ((49*n - 91)*x - (49*n - 91)*x^2)*D[P[n - 1, x], x])/(7*n - 13)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(56*n - 48 + (98*n^2 - 364*n + 240)*x + (343*n^3 - 1911*n^2 + 3206*n - 1560)*x^2)*v[i - 2] - ((49*n - 42)*x + (343*n^2 - 980*n + 588)*x^2 - (343*n^2 - 931*n + 546)*x^3)*deriv(v[i - 2], x) + (63*n - 110 + (98*n^2 - 364*n + 338)*x)*v[i - 1] + ((49*n - 91)*x - (49*n - 91)*x^2)*deriv(v[i - 1], x))/(7*n - 13)); 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.