A056241 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 4\,x - 18\,x^{2} + 14\,x^{3} - 4\,x^{4} + 3\,n - 17\,n x + 32\,n x^{2} - 22\,n x^{3} + 4\,n x^{4} - 2\,n^{2} + 18\,n^{2} x - 18\,n^{2} x^{2} + 8\,n^{2} x^{3} + n^{3} - 5\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2 - 8\,x - 30\,x^{2} + 8\,x^{3} + 14\,x^{4} - 12\,x^{5} + n - 3\,n x + 31\,n x^{2} - 6\,n x^{3} - 38\,n x^{4} + 12\,n x^{5} + 16\,n^{2} x - 2\,n^{2} x^{2} - 14\,n^{2} x^{3} + 24\,n^{2} x^{4} + n^{3} - 5\,n^{3} x + n^{3} x^{2} + 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} - 4\,x^{3} - 8\,x^{4} - 4\,x^{5} - 8\,x^{6} - 2\,n x - 2\,n x^{2} - 18\,n x^{3} - 16\,n x^{4} - 16\,n x^{5} + 2\,n^{2} x - 6\,n^{2} x^{2} - 6\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	4	5	1													10
3	7	18	9	1												35
4	11	48	49	14	1											123
5	16	105	180	105	20	1										427
6	22	201	513	501	195	27	1									1460
7	29	350	1232	1785	1162	329	35	1								4923
8	37	568	2618	5212	5027	2380	518	44	1							16405
9	46	873	5076	13182	17442	12231	4452	774	54	1						54131
10	56	1285	9165	29910	51530	49543	26715	7770	1110	65	1					177150
11	67	1826	15631	62337	134695	168476	124553	53691	12837	1540	77	1				575731
12	79	2520	25443	121330	319704	501012	481735	284790	100980	20284	2079	90	1			1860047
13	92	3393	39832	223223	701844	1339702	1613144	1240395	603460	179894	30888	2743	104	1		5978715
14	106	4473	60333	391755	1444443	3285828	4816182	4636971	2935023	1201200	306306	45591	3549	119	1	19131880

Technical details

Node id: n-c14d9c29051f3e6c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-3e4d9cd930a6e230

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:  2,   1;
  2:  4,   5,    1;
  3:  7,  18,    9,     1;
  4: 11,  48,   49,    14,     1;
  5: 16, 105,  180,   105,    20,     1;
  6: 22, 201,  513,   501,   195,    27,    1;
  7: 29, 350, 1232,  1785,  1162,   329,   35,   1;
  8: 37, 568, 2618,  5212,  5027,  2380,  518,  44,  1;
  9: 46, 873, 5076, 13182, 17442, 12231, 4452, 774, 54, 1;

%F Formula

The row polynomials satisfy (n^3-2*n^2+3*n-2-(5*n^3-18*n^2+17*n-4)*x+(4*n^3-18*n^2+32*n-18)*x^2+(8*n^2-22*n+14)*x^3+(4*n-4)*x^4)*P(n,x) = (n^3+n-2-(5*n^3-16*n^2+3*n+8)*x+(n^3-2*n^2+31*n-30)*x^2+(12*n^3-14*n^2-6*n+8)*x^3+(24*n^2-38*n+14)*x^4+(12*n-12)*x^5)*P(n-1,x) + ((2*n^2-2*n-4)*x-(6*n^2+2*n+8)*x^2-(6*n^2+18*n+4)*x^3-(8*n^2+16*n+8)*x^4-(16*n+4)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + n - 2 - (5*n^3 - 16*n^2 + 3*n + 8)*x + (n^3 - 2*n^2 + 31*n - 30)*x^2 + (12*n^3 - 14*n^2 - 6*n + 8)*x^3 + (24*n^2 - 38*n + 14)*x^4 + (12*n - 12)*x^5)*P[n - 1, x] + ((2*n^2 - 2*n - 4)*x - (6*n^2 + 2*n + 8)*x^2 - (6*n^2 + 18*n + 4)*x^3 - (8*n^2 + 16*n + 8)*x^4 - (16*n + 4)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + 3*n - 2 - (5*n^3 - 18*n^2 + 17*n - 4)*x + (4*n^3 - 18*n^2 + 32*n - 18)*x^2 + (8*n^2 - 22*n + 14)*x^3 + (4*n - 4)*x^4)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + n - 2 - (5*n^3 - 16*n^2 + 3*n + 8)*x + (n^3 - 2*n^2 + 31*n - 30)*x^2 + (12*n^3 - 14*n^2 - 6*n + 8)*x^3 + (24*n^2 - 38*n + 14)*x^4 + (12*n - 12)*x^5)*v[i - 1] + ((2*n^2 - 2*n - 4)*x - (6*n^2 + 2*n + 8)*x^2 - (6*n^2 + 18*n + 4)*x^3 - (8*n^2 + 16*n + 8)*x^4 - (16*n + 4)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + 3*n - 2 - (5*n^3 - 18*n^2 + 17*n - 4)*x + (4*n^3 - 18*n^2 + 32*n - 18)*x^2 + (8*n^2 - 22*n + 14)*x^3 + (4*n - 4)*x^4)); 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 T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

References

[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.