A112486 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0)coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(x + 4\,x^{2} + 5\,x^{3} + 2\,x^{4} + n + 3\,n x + n x^{2} - 3\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	2	9	10						21
3	6	44	105	82					237
4	24	250	946	1553	938				3711
5	120	1644	8680	22276	27953	13778			74451
6	720	12348	84680	300172	583866	593561	247210		1822557
7	5040	104544	889308	4063760	10848010	16999020	14530169	5240338	52680189

All 15 rows

Technical details

Node id: n-9c1ae989b8f28a11 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-c19223d7fb5146f9

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,        2;
  2:      2,        9,        10;
  3:      6,       44,       105,        82;
  4:     24,      250,       946,      1553,        938;
  5:    120,     1644,      8680,     22276,      27953,      13778;
  6:    720,    12348,     84680,    300172,     583866,     593561,      247210;
  7:   5040,   104544,    889308,   4063760,   10848010,   16999020,    14530169,     5240338;
  8:  40320,   986256,  10077048,  56786452,  194430720,  416285390,   546310786,   402923553,   128149802;
  9: 362880, 10265760, 123023736, 828862608, 3485617372, 9537606780, 17046350780, 19245766516, 12483933633, 3551246162;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n+(3*n+1)*x+(n+4)*x^2-(3*n-5)*x^3-(2*n-2)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*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[((n + (3*n + 1)*x + (n + 4)*x^2 - (3*n - 5)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 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] = ((n + (3*n + 1)*x + (n + 4)*x^2 - (3*n - 5)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 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 [-1, 0)
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741