A075502 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-98 + 35\,x + 6\,x^{2}\right)\,P_{n-2}(x) + \left(-49\,x + 42\,x^{2}\right)\,P_{n-2}'(x) + \left(21 - 5\,x\right)\,P_{n-1}(x) + \left(7\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 70 + 23\,x + x^{2}\)

\(P_{1}(x) = 686 + 427\,x + 45\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	70	23	1								94
1	686	427	45	1							1159
2	7546	7595	1435	74	1						16651
3	91238	136171	40425	3605	110	1					271550
4	1176490	2504243	1082851	147490	7595	153	1				4918823
5	15764966	47412547	28487865	5524701	425320	14210	203	1			97629813
6	215768266	921779915	747659395	197347794	21227241	1043406	24402	260	1		2104850680
7	2986166918	18313125691	19720913625	6871289845	988755810	66908667	2274090	39270	324	1	48949474241

All 15 rows

Technical details

Node id: n-6e894b02c6fc9b07 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8a154e29e4f61049

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:           70,            23,              1;
  1:          686,           427,             45,             1;
  2:         7546,          7595,           1435,            74,             1;
  3:        91238,        136171,          40425,          3605,           110,            1;
  4:      1176490,       2504243,        1082851,        147490,          7595,          153,           1;
  5:     15764966,      47412547,       28487865,       5524701,        425320,        14210,         203,         1;
  6:    215768266,     921779915,      747659395,     197347794,      21227241,      1043406,       24402,       260,       1;
  7:   2986166918,   18313125691,    19720913625,    6871289845,     988755810,     66908667,     2274090,     39270,     324,     1;
  8:  41564215210,  369886926563,   524396770051,  236141543330,   44110727815,   3938434731,   182156667,   4527600,   60060,   395,   1;
  9: 580204161446, 7561499233267, 14067180928185, 8069162433101, 1914626884940, 218994201580, 13217411361, 442946427, 8392230, 88165, 473, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(98-35*x-6*x^2)*P(n-2,x) - (49*x-42*x^2)*d/dx P(n-2,x) + (21-5*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 70 + 23*x + x^2, P(1,x) = 686 + 427*x + 45*x^2 + x^3.

%t Mathematica

P[0, x_] := 70 + 23*x + x^2;
P[1, x_] := 686 + 427*x + 45*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(98 - 35*x - 6*x^2)*P[n - 2, x] - (49*x - 42*x^2)*D[P[n - 2, x], x] + (21 - 5*x)*P[n - 1, x] + 7*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] = 70 + 23*x + x^2; v[2] = 686 + 427*x + 45*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(98 - 35*x - 6*x^2)*v[i - 2] - (49*x - 42*x^2)*deriv(v[i - 2], x) + (21 - 5*x)*v[i - 1] + 7*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 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).?
  - 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.