A225472 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 3\,x\right)\,P_{n-1}(x) + \left(6 + 9\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	5	3							8
3	43	57	18						118
4	557	1143	756	162					2618
5	9643	26745	27162	11988	1944				77482
6	208685	729303	1000980	674730	223560	29160			2866418
7	5419243	22848057	39543498	35965620	18137520	4811400	524880		127250218
8	164184557	810652503	1693969956	1942091802	1319567760	531528480	117573120	11022480	6590590658

All 15 rows

Technical details

Node id: n-6c1b3ca24c1db31c · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-bba9e798c16ddeb5

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:            5,             3;
   3:           43,            57,            18;
   4:          557,          1143,           756,           162;
   5:         9643,         26745,         27162,         11988,          1944;
   6:       208685,        729303,       1000980,        674730,        223560,         29160;
   7:      5419243,      22848057,      39543498,      35965620,      18137520,       4811400,        524880;
   8:    164184557,     810652503,    1693969956,    1942091802,    1319567760,     531528480,     117573120,     11022480;
   9:   5684837803,   32169328185,   78782876442,  109062293508,   93353234184,   50602572720,   16967270880,   3218564160,   264539520;
  10: 221440158125, 1412819625303, 3968143410420, 6439516900650, 6654307305240, 4541248900440, 2047094985600, 587872948320, 97615082880, 7142567040;

%F Formula

The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) + (6+9*x+3*x^2)*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[(5 + 3*x)*P[n - 1, x] + (6 + 9*x + 3*x^2)*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] = (5 + 3*x)*v[i - 1] + (6 + 9*x + 3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - 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