A103633 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(14 + 22\,x + 8\,x^{2} - 23\,n - 34\,n x - 12\,n x^{2} + 9\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} - 112\,n - 243\,n x - 178\,n x^{2} - 44\,n x^{3} + 111\,n^{2} + 227\,n^{2} x + 156\,n^{2} x^{2} + 36\,n^{2} x^{3} - 27\,n^{3} - 54\,n^{3} x - 36\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(8 + 28\,x + 36\,x^{2} + 20\,x^{3} + 4\,x^{4} - 60\,n - 178\,n x - 194\,n x^{2} - 92\,n x^{3} - 16\,n x^{4} + 36\,n^{2} + 102\,n^{2} x + 106\,n^{2} x^{2} + 48\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	3	2													5
3	5	8	3												16
4	12	26	18	4											60
5	16	50	57	28	5										156
6	36	132	189	132	45	6									540
7	44	204	387	384	210	60	7								1296
8	96	496	1080	1284	900	372	84	8							4320
9	112	688	1824	2724	2505	1452	518	104	9						9936
10	240	1600	4680	7880	8415	5910	2730	800	135	10					32400
11	272	2080	7080	14120	18265	16008	9625	3920	1035	160	11				72576
12	576	4704	17280	37680	54180	53934	37926	18840	6480	1470	198	12			233280
13	640	5856	24336	60720	101280	118950	100849	62184	27675	8670	1815	228	13		513216
14	1344	12992	57456	153888	278460	359436	340473	239624	125307	48090	13167	2436	273	14	1632960

Technical details

Node id: n-c339f708acd8421e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-97e8a3dee23710ef

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   0;
  1:   1;
  2:   3,   2;
  3:   5,   8,    3;
  4:  12,  26,   18,    4;
  5:  16,  50,   57,   28,    5;
  6:  36, 132,  189,  132,   45,    6;
  7:  44, 204,  387,  384,  210,   60,   7;
  8:  96, 496, 1080, 1284,  900,  372,  84,   8;
  9: 112, 688, 1824, 2724, 2505, 1452, 518, 104, 9;

%F Formula

The row polynomials satisfy (9*n^2-23*n+14+(12*n^2-34*n+22)*x+(4*n^2-12*n+8)*x^2)*P(n,x) = -(27*n^3-111*n^2+112*n-8+(54*n^3-227*n^2+243*n-24)*x+(36*n^3-156*n^2+178*n-24)*x^2+(8*n^3-36*n^2+44*n-8)*x^3)*P(n-1,x) + (36*n^2-60*n+8+(102*n^2-178*n+28)*x+(106*n^2-194*n+36)*x^2+(48*n^2-92*n+20)*x^3+(8*n^2-16*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 111*n^2 + 112*n - 8 + (54*n^3 - 227*n^2 + 243*n - 24)*x + (36*n^3 - 156*n^2 + 178*n - 24)*x^2 + (8*n^3 - 36*n^2 + 44*n - 8)*x^3)*P[n - 1, x] + (36*n^2 - 60*n + 8 + (102*n^2 - 178*n + 28)*x + (106*n^2 - 194*n + 36)*x^2 + (48*n^2 - 92*n + 20)*x^3 + (8*n^2 - 16*n + 4)*x^4)*D[P[n - 1, x], x])/(9*n^2 - 23*n + 14 + (12*n^2 - 34*n + 22)*x + (4*n^2 - 12*n + 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(27*n^3 - 111*n^2 + 112*n - 8 + (54*n^3 - 227*n^2 + 243*n - 24)*x + (36*n^3 - 156*n^2 + 178*n - 24)*x^2 + (8*n^3 - 36*n^2 + 44*n - 8)*x^3)*v[i - 1] + (36*n^2 - 60*n + 8 + (102*n^2 - 178*n + 28)*x + (106*n^2 - 194*n + 36)*x^2 + (48*n^2 - 92*n + 20)*x^3 + (8*n^2 - 16*n + 4)*x^4)*deriv(v[i - 1], x))/(9*n^2 - 23*n + 14 + (12*n^2 - 34*n + 22)*x + (4*n^2 - 12*n + 8)*x^2)); 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 shift_x_plus_one of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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.