A290316 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-17 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(44 - 1311\,x + 32481\,x^{2} - 22086\,x^{3} + 2592\,x^{4} - 24\,n + 1098\,n x - 38934\,n x^{2} + 16038\,n x^{3} - 972\,n x^{4} - 216\,n^{2} x + 15228\,n^{2} x^{2} - 2916\,n^{2} x^{3} - 1944\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(33\,x - 1800\,x^{2} + 6318\,x^{3} - 3807\,x^{4} + 486\,x^{5} - 18\,n x + 1566\,n x^{2} - 5022\,n x^{3} + 1458\,n x^{4} - 324\,n^{2} x^{2} + 972\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-79 + 1407\,x - 693\,x^{2} + 54\,x^{3} + 30\,n - 1098\,n x + 270\,n x^{2} + 216\,n^{2} x\right)\,P_{n-1}(x) + \left(-51\,x + 171\,x^{2} - 54\,x^{3} + 18\,n x - 54\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 7 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	7	6							13
3	49	228	90						367
4	235	4662	8694	2160					15751
5	997	55860	371196	399708	71280				899041
6	4063	525558	8636148	29874312	21739428	2993760			63773269
7	16345	4373604	146947662	1170865800	2551341564	1370228400	152681760		5396455135
8	65491	33998550	2092102290	32078842560	153632231820	234575876232	98362803600	9160905600	529936826143

All 15 rows

Technical details

Node id: n-75904bb0bb525186 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-cb98ad15abc25bd9

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:       7,          6;
   3:      49,        228,           90;
   4:     235,       4662,         8694,           2160;
   5:     997,      55860,       371196,         399708,           71280;
   6:    4063,     525558,      8636148,       29874312,        21739428,          2993760;
   7:   16345,    4373604,    146947662,     1170865800,      2551341564,       1370228400,        152681760;
   8:   65491,   33998550,   2092102290,    32078842560,    153632231820,     234575876232,      98362803600,       9160905600;
   9:  262093,  253904052,  26680731840,   707403008340,   6273660789540,   20342762406576,   23291048663784,    7928804037600,    632102486400;
  10: 1048519, 1850452182, 316684918512, 13541266947960, 200486987121960, 1171238676374808, 2772692349214104, 2494472439737664, 709401128349600, 49303993939200;

%F Formula

The row polynomials satisfy (6*n-17+6*x)*P(n,x) = -(24*n-44+(216*n^2-1098*n+1311)*x+(1944*n^3-15228*n^2+38934*n-32481)*x^2+(2916*n^2-16038*n+22086)*x^3+(972*n-2592)*x^4)*P(n-2,x) - ((18*n-33)*x+(324*n^2-1566*n+1800)*x^2-(972*n^2-5022*n+6318)*x^3-(1458*n-3807)*x^4-486*x^5)*d/dx P(n-2,x) + (30*n-79+(216*n^2-1098*n+1407)*x+(270*n-693)*x^2+54*x^3)*P(n-1,x) + ((18*n-51)*x-(54*n-171)*x^2-54*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 6*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 7 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n - 44 + (216*n^2 - 1098*n + 1311)*x + (1944*n^3 - 15228*n^2 + 38934*n - 32481)*x^2 + (2916*n^2 - 16038*n + 22086)*x^3 + (972*n - 2592)*x^4)*P[n - 2, x] - ((18*n - 33)*x + (324*n^2 - 1566*n + 1800)*x^2 - (972*n^2 - 5022*n + 6318)*x^3 - (1458*n - 3807)*x^4 - 486*x^5)*D[P[n - 2, x], x] + (30*n - 79 + (216*n^2 - 1098*n + 1407)*x + (270*n - 693)*x^2 + 54*x^3)*P[n - 1, x] + ((18*n - 51)*x - (54*n - 171)*x^2 - 54*x^3)*D[P[n - 1, x], x])/(6*n - 17 + 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] = 7 + 6*x; for(n = 3, N, my(i = n); v[i] = (-(24*n - 44 + (216*n^2 - 1098*n + 1311)*x + (1944*n^3 - 15228*n^2 + 38934*n - 32481)*x^2 + (2916*n^2 - 16038*n + 22086)*x^3 + (972*n - 2592)*x^4)*v[i - 2] - ((18*n - 33)*x + (324*n^2 - 1566*n + 1800)*x^2 - (972*n^2 - 5022*n + 6318)*x^3 - (1458*n - 3807)*x^4 - 486*x^5)*deriv(v[i - 2], x) + (30*n - 79 + (216*n^2 - 1098*n + 1407)*x + (270*n - 693)*x^2 + 54*x^3)*v[i - 1] + ((18*n - 51)*x - (54*n - 171)*x^2 - 54*x^3)*deriv(v[i - 1], x))/(6*n - 17 + 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.