A257626 → D

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-54 + 36\,x - 54\,n + 9\,n x^{2} - 36\,n^{2} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-18\,x + 18\,x^{2} - 18\,n x + 9\,n x^{2} + 9\,n x^{3} - 9\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(6 + 15\,n + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 6\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	6														6
2	108	72													180
3	1404	2808	648												4860
4	15876	67392	47628	5184											136080
5	166212	1284984	1927476	664848	38880										4082400
6	1659204	21403440	57824280	42806880	8296020	279936									132269760
7	16052580	326527848	1445257080	1927009440	816319620	96315480	1959552								4629441600
8	152031492	4688858016	31964483412	69372339840	53274139020	14066574048	1064220444	13436928							174596083200
9	1418671908	64475996760	648266139756	2143707652656	2679634565820	1296532279512	225665988660	11349375264	90699264						7071141369600
10	13100611140	858832275600	12335269965120	59331254220000	112544651764440	88996881330000	28782296585280	3435329102400	117905500260	604661760					306416126016000
11	120082282596	11170627642440	223676501878800	1512032294704320	4143375313653240	4972050376383888	2646056515732560	596470671676800	50267824390980	1200822825960	3990767616				14156425021939200
12	1094889628548	142693456056192	3908093596482780	36163641379829760	138041971532988840	238658418066426624	193258760146184376	72327282759659520	11724280789448340	713467280280960	12043785914028	26121388032			694951773804288000
13	9945431515044	1797722863701048	66326850574276068	822901663082178000	4255017953936381640	10200361911263077392	11900422229806923624	6808028726298210624	1851528741934900500	221089501914253560	9887475750355764	119345178180528	169789022208		36137492237822976000
14	90096614866116	22408090611676272	1100069546140652328	17995918763044455264	123300070634803856940	397989332212023596448	642622800409573875696	530652442949364795264	221940127142646942492	44989796907611138160	4033588335849058536	134448543670057632	1171255993259508	1097098297344	1984782265985046528000

Technical details

Node id: n-5f3bc25b4b68067b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-96547810142a253d

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:          6;
  2:        108,          72;
  3:       1404,        2808,          648;
  4:      15876,       67392,        47628,          5184;
  5:     166212,     1284984,      1927476,        664848,         38880;
  6:    1659204,    21403440,     57824280,      42806880,       8296020,        279936;
  7:   16052580,   326527848,   1445257080,    1927009440,     816319620,      96315480,      1959552;
  8:  152031492,  4688858016,  31964483412,   69372339840,   53274139020,   14066574048,   1064220444,    13436928;
  9: 1418671908, 64475996760, 648266139756, 2143707652656, 2679634565820, 1296532279512, 225665988660, 11349375264, 90699264;

%F Formula

The row polynomials satisfy n*P(n,x) = -(54*n+54+(36*n^2-36)*x+(9*n^3-9*n)*x^2)*P(n-2,x) - ((18*n+18)*x+(9*n^2-9*n-18)*x^2-(9*n^2+9*n)*x^3)*d/dx P(n-2,x) + (15*n+6+6*n^2*x)*P(n-1,x) + (3*n*x-3*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(54*n + 54 + (36*n^2 - 36)*x + (9*n^3 - 9*n)*x^2)*P[n - 2, x] - ((18*n + 18)*x + (9*n^2 - 9*n - 18)*x^2 - (9*n^2 + 9*n)*x^3)*D[P[n - 2, x], x] + (15*n + 6 + 6*n^2*x)*P[n - 1, x] + (3*n*x - 3*n*x^2)*D[P[n - 1, x], x])/(n)]];
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] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(54*n + 54 + (36*n^2 - 36)*x + (9*n^3 - 9*n)*x^2)*v[i - 2] - ((18*n + 18)*x + (9*n^2 - 9*n - 18)*x^2 - (9*n^2 + 9*n)*x^3)*deriv(v[i - 2], x) + (15*n + 6 + 6*n^2*x)*v[i - 1] + (3*n*x - 3*n*x^2)*deriv(v[i - 1], x))/(n)); 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) = 3*x + 6.?
  - 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) = 3*x + 6.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.