A290595 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-10 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(280 + 1048\,x - 635\,x^{2} - 549\,x^{3} + 126\,x^{4} - 393\,n - 1137\,n x + 144\,n x^{2} + 432\,n x^{3} - 54\,n x^{4} + 180\,n^{2} + 423\,n^{2} x + 108\,n^{2} x^{2} - 81\,n^{2} x^{3} - 27\,n^{3} - 54\,n^{3} x - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-168\,x + 102\,x^{2} + 309\,x^{3} - 297\,x^{4} + 54\,x^{5} + 135\,n x - 36\,n x^{2} - 180\,n x^{3} + 81\,n x^{4} - 27\,n^{2} x + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(76 + 110\,x - 141\,x^{2} + 18\,x^{3} - 81\,n - 75\,n x + 45\,n x^{2} + 18\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x + 48\,x^{2} - 18\,x^{3} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	2							5
3	23	27	4						54
4	250	516	171	8					945
5	3474	11450	6789	951	16				22680
6	58664	282156	271180	71878	4995	32			688905
7	1165576	7644912	10956816	4785412	681519	25551	64		25259850
8	26632752	226615016	453098292	299180520	71921390	6070368	129099	128	1083647565

All 15 rows

Technical details

Node id: n-e6ca5cf785756a3f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8449ee3a266702aa

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 -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:           1;
   2:           3,            2;
   3:          23,           27,            4;
   4:         250,          516,          171,             8;
   5:        3474,        11450,         6789,           951,           16;
   6:       58664,       282156,       271180,         71878,         4995,           32;
   7:     1165576,      7644912,     10956816,       4785412,       681519,        25551,          64;
   8:    26632752,    226615016,    453098292,     299180520,     71921390,      6070368,      129099,       128;
   9:   687950000,   7315140432,  19382702668,   18109844564,   6667956150,    980938510,    52062953,    648567,     256;
  10: 19822519360, 255937349952, 863843561784, 1084479861048, 571396220388, 130661739900, 12567234288, 435999546, 3249747, 512;

%F Formula

The row polynomials satisfy (3*n-10+6*x)*P(n,x) = -(27*n^3-180*n^2+393*n-280+(54*n^3-423*n^2+1137*n-1048)*x+(27*n^3-108*n^2-144*n+635)*x^2+(81*n^2-432*n+549)*x^3+(54*n-126)*x^4)*P(n-2,x) - ((27*n^2-135*n+168)*x+(36*n-102)*x^2-(27*n^2-180*n+309)*x^3-(81*n-297)*x^4-54*x^5)*d/dx P(n-2,x) + (18*n^2-81*n+76+(18*n^2-75*n+110)*x+(45*n-141)*x^2+18*x^3)*P(n-1,x) + ((9*n-30)*x-(9*n-48)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 180*n^2 + 393*n - 280 + (54*n^3 - 423*n^2 + 1137*n - 1048)*x + (27*n^3 - 108*n^2 - 144*n + 635)*x^2 + (81*n^2 - 432*n + 549)*x^3 + (54*n - 126)*x^4)*P[n - 2, x] - ((27*n^2 - 135*n + 168)*x + (36*n - 102)*x^2 - (27*n^2 - 180*n + 309)*x^3 - (81*n - 297)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (18*n^2 - 81*n + 76 + (18*n^2 - 75*n + 110)*x + (45*n - 141)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 30)*x - (9*n - 48)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 10 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(27*n^3 - 180*n^2 + 393*n - 280 + (54*n^3 - 423*n^2 + 1137*n - 1048)*x + (27*n^3 - 108*n^2 - 144*n + 635)*x^2 + (81*n^2 - 432*n + 549)*x^3 + (54*n - 126)*x^4)*v[i - 2] - ((27*n^2 - 135*n + 168)*x + (36*n - 102)*x^2 - (27*n^2 - 180*n + 309)*x^3 - (81*n - 297)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (18*n^2 - 81*n + 76 + (18*n^2 - 75*n + 110)*x + (45*n - 141)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 30)*x - (9*n - 48)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 10 + 6*x)); 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 T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - 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.