A112493 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 9\,x^{2} + 10\,x^{3} + 4\,x^{4} - 2\,n x^{2} - 4\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	2														3
3	1	6	8													15
4	1	14	50	52												117
5	1	30	214	552	472											1269
6	1	62	784	3740	7576	5504										17667
7	1	126	2652	20736	74316	124128	78416									300375
8	1	254	8574	102900	578460	1660776	2361824	1320064								6032853
9	1	510	26978	477144	3938460	17239896	41295200	51153888	25637824							139769901
10	1	1022	83468	2117260	24617148	153806376	550636952	1131695728	1242099040	564275648						3669332643
11	1	2046	255496	9120432	145207048	1242071208	6222995576	18835853856	33904071392	33410548608	13879795712					107649921375
12	1	4094	776698	38488868	822272056	9357639872	62933694992	262726030288	688577316608	1102596219136	985967376896	377332365568				3490352185077
13	1	8190	2350542	160087176	4520757264	67041826128	588103933984	3248190463680	11610557596176	26827164180864	38689467601152	31668559570944	11234698041088			123938466417189
14	1	16382	7092552	658972860	24317734416	462659989296	5184106465840	36809016823328	172146151822416	537664234672416	1110687444088320	1457122693003008	1099692789112576	363581406419456		4783375486212867
15	1	32766	21359540	2692238688	128681337396	3103228760400	43716245859424	390751704074688	2323345262816528	9410734092227040	26088363172247424	48726156649483008	58631175849038848	41049974904138240	12707452084972544	199374904588586535

Technical details

Node id: n-27eab39bb9297ea0 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-36a7872f2b6a1747

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,    2;
   3: 1,    6,     8;
   4: 1,   14,    50,      52;
   5: 1,   30,   214,     552,      472;
   6: 1,   62,   784,    3740,     7576,      5504;
   7: 1,  126,  2652,   20736,    74316,    124128,     78416;
   8: 1,  254,  8574,  102900,   578460,   1660776,   2361824,    1320064;
   9: 1,  510, 26978,  477144,  3938460,  17239896,  41295200,   51153888,   25637824;
  10: 1, 1022, 83468, 2117260, 24617148, 153806376, 550636952, 1131695728, 1242099040, 564275648;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x-(2*n-9)*x^2-(4*n-10)*x^3-(2*n-4)*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 >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 4*x - (2*n - 9)*x^2 - (4*n - 10)*x^3 - (2*n - 4)*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, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 4*x - (2*n - 9)*x^2 - (4*n - 10)*x^3 - (2*n - 4)*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, 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, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201…?
  - 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