A075505 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-200 + 80\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-100\,x + 90\,x^{2}\right)\,P_{n-2}'(x) + \left(30 - 8\,x\right)\,P_{n-1}(x) + \left(10\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 130 + 32\,x + x^{2}\)

\(P_{1}(x) = 1700 + 820\,x + 63\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	130	32	1								163
1	1700	820	63	1							2584
2	25000	20000	2800	104	1						47905
3	410000	490000	109500	7100	155	1					1016756
4	7300000	12320000	4060000	406000	15050	216	1				24101267
5	137000000	320200000	147630000	21210000	1183000	28280	287	1			627251568
6	2650000000	8600000000	5356000000	1055040000	82740000	2923200	48720	368	1		17746752289
7	52100000000	237700000000	195615000000	51115000000	5393850000	263550000	6405000	78600	459	1	542193884060

All 15 rows

Technical details

Node id: n-5a9335c585a9eb7b · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2d0fde8c3e0130d9

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 2)
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:            130,              32,               1;
  1:           1700,             820,              63,               1;
  2:          25000,           20000,            2800,             104,              1;
  3:         410000,          490000,          109500,            7100,            155,             1;
  4:        7300000,        12320000,         4060000,          406000,          15050,           216,           1;
  5:      137000000,       320200000,       147630000,        21210000,        1183000,         28280,         287,          1;
  6:     2650000000,      8600000000,      5356000000,      1055040000,       82740000,       2923200,       48720,        368,        1;
  7:    52100000000,    237700000000,    195615000000,     51115000000,     5393850000,     263550000,     6405000,      78600,      459,      1;
  8:  1033000000000,   6723200000000,   7222600000000,   2444420000000,   336473500000,   21787920000,   723030000,   12804000,   120450,    560,   1;
  9: 20570000000000, 193522000000000, 269976300000000, 116326100000000, 20412260000000, 1700089600000, 73871490000, 1768470000, 23809500, 177100, 671, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(200-80*x-9*x^2)*P(n-2,x) - (100*x-90*x^2)*d/dx P(n-2,x) + (30-8*x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 130 + 32*x + x^2, P(1,x) = 1700 + 820*x + 63*x^2 + x^3.

%t Mathematica

P[0, x_] := 130 + 32*x + x^2;
P[1, x_] := 1700 + 820*x + 63*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(200 - 80*x - 9*x^2)*P[n - 2, x] - (100*x - 90*x^2)*D[P[n - 2, x], x] + (30 - 8*x)*P[n - 1, x] + 10*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 130 + 32*x + x^2; v[2] = 1700 + 820*x + 63*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(200 - 80*x - 9*x^2)*v[i - 2] - (100*x - 90*x^2)*deriv(v[i - 2], x) + (30 - 8*x)*v[i - 1] + 10*x*deriv(v[i - 1], x)); 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 Stirling2 triangle with scaled diagonals (powers of 10).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.