A112486 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in [−2, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	10	11	3						24
3	82	141	80	15					318
4	938	2199	1915	735	105				5892
5	13778	40937	48244	28210	8190	945			140304
6	247210	889699	1324211	1043882	459900	107415	10395		4082712
7	5240338	22152197	39865104	39606385	23470370	8298675	1621620	135135	140389824

All 15 rows

Technical details

Node id: n-6e68ecd44a6b5992 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-010a81a2cff13557

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:         10,          11,           3;
  3:         82,         141,          80,          15;
  4:        938,        2199,        1915,         735,         105;
  5:      13778,       40937,       48244,       28210,        8190,         945;
  6:     247210,      889699,     1324211,     1043882,      459900,      107415,       10395;
  7:    5240338,    22152197,    39865104,    39606385,    23470370,     8298675,     1621620,     135135;
  8:  128149802,   622274863,  1314040371,  1576573625,  1175827345,   558362805,   164909745,   27702675,   2027025;
  9: 3551246162, 19477281825, 47219159284, 66428550716, 59777331900, 35690644990, 14141667540, 3586482900, 528377850, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+n*x)*P(n-1,x) + (2+3*x+x^2)*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[(2*n + n*x)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], x]];
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] = (2*n + n*x)*v[i - 1] + (2 + 3*x + x^2)*deriv(v[i - 1], x)); 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 [-2, -1)
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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