A308737 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x + x^{2} - n + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-11 - 8\,x - 4\,x^{2} + 4\,x^{3} - x^{4} + 7\,n - n x + 10\,n x^{2} - 3\,n x^{3} + n x^{4} + 7\,n^{2} x - 3\,n^{2} x^{2} + 2\,n^{2} x^{3} - n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-3 - 5\,x - 4\,x^{2} - 2\,x^{3} - x^{4} - x^{5} + n - n x - n x^{2} - n x^{3} - 2\,n x^{4} + n^{2} x - n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	9	22	7													38
3	18	79	76	15												188
4	35	226	378	218	31											888
5	68	583	1404	1394	568	63										4080
6	133	1422	4485	6380	4455	1398	127									18400
7	262	3351	13080	24305	24270	13017	3316	255								81856
8	519	7714	35896	82446	107450	82334	35784	7666	511							360320
9	1032	17455	94312	258132	415688	415562	257880	94132	17392	1023						1572608
10	2057	38974	239787	762072	1462314	1806084	1461894	761592	239517	38894	2047					6815232
11	4106	86095	594180	2150835	4792620	7053102	7052640	4791630	2150010	593795	85996	4095				29359104
12	8203	188514	1442166	5858050	14869305	25411716	30276708	25410132	14867325	5856730	1441638	188394	8191			125827072
13	16396	409719	3441156	15500518	44156640	85967937	118947048	118945332	85964076	44153065	15498516	3440454	409576	16383		536866816
14	32781	884878	8094385	40044420	126520537	276466762	435782061	506065560	435776055	276458754	126514531	40041508	8093475	884710	32767	2281693184

Technical details

Node id: n-7eeb625c0c33664b · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0ac9bf5c1d579c84

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:    4,     3;
  2:    9,    22,     7;
  3:   18,    79,    76,     15;
  4:   35,   226,   378,    218,     31;
  5:   68,   583,  1404,   1394,    568,     63;
  6:  133,  1422,  4485,   6380,   4455,   1398,    127;
  7:  262,  3351, 13080,  24305,  24270,  13017,   3316,   255;
  8:  519,  7714, 35896,  82446, 107450,  82334,  35784,  7666,   511;
  9: 1032, 17455, 94312, 258132, 415688, 415562, 257880, 94132, 17392, 1023;

%F Formula

The row polynomials satisfy (n^2-n-1+(2*n-2)*x+x^2)*P(n,x) = (7*n-11-(n^3-7*n^2+n+8)*x+(n^3-3*n^2+10*n-4)*x^2+(2*n^2-3*n+4)*x^3+(n-1)*x^4)*P(n-1,x) + (n-3+(n^2-n-5)*x-(n+4)*x^2-(n^2+n+2)*x^3-(2*n+1)*x^4-x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 11 - (n^3 - 7*n^2 + n + 8)*x + (n^3 - 3*n^2 + 10*n - 4)*x^2 + (2*n^2 - 3*n + 4)*x^3 + (n - 1)*x^4)*P[n - 1, x] + (n - 3 + (n^2 - n - 5)*x - (n + 4)*x^2 - (n^2 + n + 2)*x^3 - (2*n + 1)*x^4 - x^5)*D[P[n - 1, x], x])/(n^2 - n - 1 + (2*n - 2)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((7*n - 11 - (n^3 - 7*n^2 + n + 8)*x + (n^3 - 3*n^2 + 10*n - 4)*x^2 + (2*n^2 - 3*n + 4)*x^3 + (n - 1)*x^4)*v[i - 1] + (n - 3 + (n^2 - n - 5)*x - (n + 4)*x^2 - (n^2 + n + 2)*x^3 - (2*n + 1)*x^4 - x^5)*deriv(v[i - 1], x))/(n^2 - n - 1 + (2*n - 2)*x + 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 Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.