A359363 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-18\,x - 12\,x^{2} + 6\,x^{3} - 6\,n - 33\,n x + 8\,n x^{2} + 5\,n x^{3} - 5\,n^{2} + 13\,n^{2} x^{2} + n^{2} x^{3} + 5\,n^{3} + 12\,n^{3} x + 3\,n^{3} x^{2} + 5\,n^{4} + 3\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(18\,x + 21\,x^{2} - 3\,x^{4} - 6\,n + 6\,n x - 20\,n x^{2} - 12\,n x^{3} + 5\,n x^{4} - 5\,n^{2} - 47\,n^{2} x - 55\,n^{2} x^{2} + 6\,n^{2} x^{3} + 10\,n^{2} x^{4} + 5\,n^{3} - 31\,n^{3} x + 30\,n^{3} x^{3} + 5\,n^{4} + 8\,n^{4} x + 30\,n^{4} x^{2} + n^{5} + 10\,n^{5} x\right)\,P_{n-1}(x) + \left(-36\,x - 69\,x^{2} - 27\,x^{3} + 9\,x^{4} + 3\,x^{5} - 15\,n x + 37\,n x^{2} + 79\,n x^{3} + 12\,n x^{4} - 15\,n x^{5} + 27\,n^{2} x + 83\,n^{2} x^{2} + 17\,n^{2} x^{3} - 45\,n^{2} x^{4} + 21\,n^{3} x - 45\,n^{3} x^{3} + 3\,n^{4} x - 15\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(-12\,x^{2} - 24\,x^{3} - 6\,x^{4} + 12\,x^{5} + 6\,x^{6} - 4\,n x^{2} + 8\,n x^{3} + 30\,n x^{4} + 18\,n x^{5} + 10\,n^{2} x^{2} + 28\,n^{2} x^{3} + 18\,n^{2} x^{4} + 6\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	3	1						5
3	1	9	7	1					18
4	1	21	40	14	1				77
5	1	41	170	130	25	1			368
6	1	71	560	875	350	41	1		1899
7	1	113	1526	4410	3430	826	63	1	10370

All 15 rows

Technical details

Node id: n-b0113a98618fdee8 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d997120effc7e338

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: 1,   1;
  2: 1,   3,    1;
  3: 1,   9,    7,     1;
  4: 1,  21,   40,    14,      1;
  5: 1,  41,  170,   130,     25,      1;
  6: 1,  71,  560,   875,    350,     41,     1;
  7: 1, 113, 1526,  4410,   3430,    826,    63,    1;
  8: 1, 169, 3612, 17640,  24696,  11172,  1764,   92,   1;
  9: 1, 241, 7680, 58968, 137592, 109368, 31752, 3480, 129, 1;

%F Formula

The row polynomials satisfy (n^5+5*n^4+5*n^3-5*n^2-6*n+(3*n^4+12*n^3-33*n-18)*x+(3*n^3+13*n^2+8*n-12)*x^2+(n^2+5*n+6)*x^3)*P(n,x) = (n^5+5*n^4+5*n^3-5*n^2-6*n+(10*n^5+8*n^4-31*n^3-47*n^2+6*n+18)*x+(30*n^4-55*n^2-20*n+21)*x^2+(30*n^3+6*n^2-12*n)*x^3+(10*n^2+5*n-3)*x^4)*P(n-1,x) + ((3*n^4+21*n^3+27*n^2-15*n-36)*x-(15*n^4-83*n^2-37*n+69)*x^2-(45*n^3-17*n^2-79*n+27)*x^3-(45*n^2-12*n-9)*x^4-(15*n-3)*x^5)*d/dx P(n-1,x) + ((6*n^3+10*n^2-4*n-12)*x^2+(6*n^3+28*n^2+8*n-24)*x^3+(18*n^2+30*n-6)*x^4+(18*n+12)*x^5+6*x^6)*d^2/dx^2 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[((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 8*n^4 - 31*n^3 - 47*n^2 + 6*n + 18)*x + (30*n^4 - 55*n^2 - 20*n + 21)*x^2 + (30*n^3 + 6*n^2 - 12*n)*x^3 + (10*n^2 + 5*n - 3)*x^4)*P[n - 1, x] + ((3*n^4 + 21*n^3 + 27*n^2 - 15*n - 36)*x - (15*n^4 - 83*n^2 - 37*n + 69)*x^2 - (45*n^3 - 17*n^2 - 79*n + 27)*x^3 - (45*n^2 - 12*n - 9)*x^4 - (15*n - 3)*x^5)*D[P[n - 1, x], x] + ((6*n^3 + 10*n^2 - 4*n - 12)*x^2 + (6*n^3 + 28*n^2 + 8*n - 24)*x^3 + (18*n^2 + 30*n - 6)*x^4 + (18*n + 12)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (3*n^4 + 12*n^3 - 33*n - 18)*x + (3*n^3 + 13*n^2 + 8*n - 12)*x^2 + (n^2 + 5*n + 6)*x^3)]];
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] = ((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 8*n^4 - 31*n^3 - 47*n^2 + 6*n + 18)*x + (30*n^4 - 55*n^2 - 20*n + 21)*x^2 + (30*n^3 + 6*n^2 - 12*n)*x^3 + (10*n^2 + 5*n - 3)*x^4)*v[i - 1] + ((3*n^4 + 21*n^3 + 27*n^2 - 15*n - 36)*x - (15*n^4 - 83*n^2 - 37*n + 69)*x^2 - (45*n^3 - 17*n^2 - 79*n + 27)*x^3 - (45*n^2 - 12*n - 9)*x^4 - (15*n - 3)*x^5)*deriv(v[i - 1], x) + ((6*n^3 + 10*n^2 - 4*n - 12)*x^2 + (6*n^3 + 28*n^2 + 8*n - 24)*x^3 + (18*n^2 + 30*n - 6)*x^4 + (18*n + 12)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (3*n^4 + 12*n^3 - 33*n - 18)*x + (3*n^3 + 13*n^2 + 8*n - 12)*x^2 + (n^2 + 5*n + 6)*x^3)); 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. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 …?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[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.