A166979 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 + 7\,x + 8\,x^{2}\right)\,P_{n-2}(x) + \left(-9\,x + 72\,x^{2}\right)\,P_{n-2}'(x) + \left(11 - 7\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 12 + 13\,x + x^{2}\)

\(P_{2}(x) = 112 + 171\,x + 33\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	12	13	1								26
2	112	171	33	1							317
3	1112	2473	855	62	1						4503
4	11112	39211	20965	2685	100	1					74074
5	111112	667233	513831	101870	6525	147	1				1400719
6	1111112	11899651	12830013	3654301	355460	13482	203	1			29864223
7	11111112	218315593	327985855	128210502	17654021	998522	24906	268	1		704300780
8	111111112	4070218491	8573101605	4473418125	833463540	65692683	2407818	42390	342	1	18129456107

All 15 rows

Technical details

Node id: n-2c4a421bbeed7ecb · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-87f160794a2bbf3d

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 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:          12,            13,             1;
   2:         112,           171,            33,             1;
   3:        1112,          2473,           855,            62,             1;
   4:       11112,         39211,         20965,          2685,           100,            1;
   5:      111112,        667233,        513831,        101870,          6525,          147,           1;
   6:     1111112,      11899651,      12830013,       3654301,        355460,        13482,         203,         1;
   7:    11111112,     218315593,     327985855,     128210502,      17654021,       998522,       24906,       268,        1;
   8:   111111112,    4070218491,    8573101605,    4473418125,     833463540,     65692683,     2407818,     42390,      342,      1;
   9:  1111111112,   76556373553,  228280633911,  156506777790,   38261603325,   4023246969,   202755147,   5184480,    67770,    425,   1;
  10: 11111111112, 1446793319731, 6166633073293, 5510876738301, 1730565643620, 235516598724, 15639178401, 543844587, 10229340, 103125, 517, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(10-7*x-8*x^2)*P(n-2,x) - (9*x-72*x^2)*d/dx P(n-2,x) + (11-7*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 13*x + x^2, P(2,x) = 112 + 171*x + 33*x^2 + x^3.

%t Mathematica

P[1, x_] := 12 + 13*x + x^2;
P[2, x_] := 112 + 171*x + 33*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(10 - 7*x - 8*x^2)*P[n - 2, x] - (9*x - 72*x^2)*D[P[n - 2, x], x] + (11 - 7*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 12 + 13*x + x^2; v[2] = 112 + 171*x + 33*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = -(10 - 7*x - 8*x^2)*v[i - 2] - (9*x - 72*x^2)*deriv(v[i - 2], x) + (11 - 7*x)*v[i - 1] + 9*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, 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: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m…?
  - 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.