A094262 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 7\,x - 8\,x^{2} - 2\,x^{3} + 2\,x^{4} + x^{5}\right)\,P_{n-2}(x) + \left(-x - 3\,x^{2} - 2\,x^{3} + 2\,x^{4} + 3\,x^{5} + x^{6}\right)\,P_{n-2}'(x) + \left(3 + 7\,x + 3\,x^{2} - x^{3}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 4\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4	1													8
2	7	30	42	22	3											104
3	15	136	433	648	481	160	15									1888
4	31	510	2910	8210	12960	11826	6010	1470	105							44032
5	63	1744	16021	73020	193070	317588	332634	218596	84595	16380	945					1254656
6	127	5670	79002	537222	2135133	5423292	9212700	10654092	8343657	4285918	1340010	214830	10395			42242048
7	255	17896	364393	3523968	19800291	71684760	177205581	308843328	384923613	342742296	214277035	90354880	23704065	3243240	135135	1640820736

All 15 rows

Technical details

Node id: n-3d2f72d29c0e6cb2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-75519f96f3248de5

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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:    1;
  1:    3,      4,       1;
  2:    7,     30,      42,        22,          3;
  3:   15,    136,     433,       648,        481,        160,          15;
  4:   31,    510,    2910,      8210,      12960,      11826,        6010,         1470,          105;
  5:   63,   1744,   16021,     73020,     193070,     317588,      332634,       218596,        84595,        16380,          945;
  6:  127,   5670,   79002,    537222,    2135133,    5423292,     9212700,     10654092,      8343657,      4285918,      1340010,       214830,        10395;
  7:  255,  17896,  364393,   3523968,   19800291,   71684760,   177205581,    308843328,    384923613,    342742296,    214277035,     90354880,     23704065,      3243240,       135135;
  8:  511,  55470, 1610070,  21453370,  163863570,  805396662,  2724185310,   6604150770,  11752457040,  15532188490,  15262069026,  11040969390,   5743195150,   2054021970,    464233770,    55405350,    2027025;
  9: 1023, 169984, 6912541, 124115940, 1254376240, 8101721116, 36040535448, 115738941732, 276500281830, 500602443900, 693602097178, 737001842764, 597227576616, 363697159540, 161963161360, 50290700460, 9984449475, 1056755700, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = -(2+7*x+8*x^2+2*x^3-2*x^4-x^5)*P(n-2,x) - (x+3*x^2+2*x^3-2*x^4-3*x^5-x^6)*d/dx P(n-2,x) + (3+7*x+3*x^2-x^3)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(2 + 7*x + 8*x^2 + 2*x^3 - 2*x^4 - x^5)*P[n - 2, x] - (x + 3*x^2 + 2*x^3 - 2*x^4 - 3*x^5 - x^6)*D[P[n - 2, x], x] + (3 + 7*x + 3*x^2 - x^3)*P[n - 1, x] + (x + 2*x^2 + x^3)*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] = 1; v[2] = 3 + 4*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 7*x + 8*x^2 + 2*x^3 - 2*x^4 - x^5)*v[i - 2] - (x + 3*x^2 + 2*x^3 - 2*x^4 - 3*x^5 - x^6)*deriv(v[i - 2], x) + (3 + 7*x + 3*x^2 - x^3)*v[i - 1] + (x + 2*x^2 + x^3)*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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.