A131689 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	2	5	3						10
4	6	24	31	13					74
5	24	132	266	233	75				730
6	120	840	2310	3120	2071	541			9002
7	720	6120	21360	39180	39842	21305	4683		133210
8	5040	50400	213360	495600	682206	556584	249271	47293	2299754

All 15 rows

Technical details

Node id: n-efc52f864f8006b6 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-eef4c9e321b3f244

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      1,       1;
   3:      2,       5,        3;
   4:      6,      24,       31,       13;
   5:     24,     132,      266,      233,        75;
   6:    120,     840,     2310,     3120,      2071,       541;
   7:    720,    6120,    21360,    39180,     39842,     21305,      4683;
   8:   5040,   50400,   213360,   495600,    682206,    556584,    249271,     47293;
   9:  40320,  463680,  2308320,  6496560,  11305224,  12456612,   8487866,   3270713,   545835;
  10: 362880, 4717440, 27004320, 89328960, 188170920, 261759960, 240471510, 140696160, 47580151, 7087261;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n - 1 + (3*n - 5)*x + (2*n - 4)*x^2)*v[i - 1] - (x + 3*x^2 + 2*x^3)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 shift_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for …?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.?
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.?
    
    ↑ Ma–Wang interval theorem (above)

Related ?

Same fingerprint ?

A131689 → x ↦ x+1

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