A225473 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 21\,x + 49\,x^{2} + 47\,x^{3} + 16\,x^{4} - 4\,n x - 16\,n x^{2} - 20\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 20\,x^{2} + 36\,x^{3} + 28\,x^{4} + 8\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	7							10
3	9	58	81						148
4	27	397	1385	1399					3208
5	81	2644	17366	40948	32289				93328
6	243	17779	196366	811254	1441231	931687			3398560
7	729	121294	2132527	13759908	41329447	58891822	32259441		148495168
8	2187	836665	22816335	216474549	978456881	2305486971	2742082101	1303134679	7569290368

All 15 rows

Technical details

Node id: n-7cfc47df0195bedb · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-0833365c0b4ddd0e

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:     3,        7;
   3:     9,       58,         81;
   4:    27,      397,       1385,        1399;
   5:    81,     2644,      17366,       40948,        32289;
   6:   243,    17779,     196366,      811254,      1441231,        931687;
   7:   729,   121294,    2132527,    13759908,     41329447,      58891822,      32259441;
   8:  2187,   836665,   22816335,   216474549,    978456881,    2305486971,    2742082101,    1303134679;
   9:  6561,  5810728,  243327228,  3275407320,  20925145574,   72517350552,  140467243452,  143355570664,   60160827969;
  10: 19683, 40511071, 2600640140, 48562381804, 421927539482, 2018161118098, 5661118184156, 9305566719580, 8316131752859, 3124574220487;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3-(4*n-21)*x-(16*n-49)*x^2-(20*n-47)*x^3-(8*n-16)*x^4)*P(n-1,x) + (4*x+20*x^2+36*x^3+28*x^4+8*x^5)*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[Together[((3 - (4*n - 21)*x - (16*n - 49)*x^2 - (20*n - 47)*x^3 - (8*n - 16)*x^4)*P[n - 1, x] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((3 - (4*n - 21)*x - (16*n - 49)*x^2 - (20*n - 47)*x^3 - (8*n - 16)*x^4)*v[i - 1] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*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, 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 Triangle read by rows, k!*S_4(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