A112486 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	2	9	10													21
3	6	44	105	82												237
4	24	250	946	1553	938											3711
5	120	1644	8680	22276	27953	13778										74451
6	720	12348	84680	300172	583866	593561	247210									1822557
7	5040	104544	889308	4063760	10848010	16999020	14530169	5240338								52680189
8	40320	986256	10077048	56786452	194430720	416285390	546310786	402923553	128149802							1755990327
9	362880	10265760	123023736	828862608	3485617372	9537606780	17046350780	19245766516	12483933633	3551246162						66313036227
10	3628800	116915040	1613473632	12702365256	63653563064	213286949960	485863215040	745221349152	738297229226	427428288169	109979486890					2798166464229
11	39916800	1446526080	22656306816	204771168240	1195737538224	4762443599312	13254579547832	25857011236336	34733046156990	30655937920604	16026083546377	3764281873042				130478035336653
12	479001600	19323757440	339454543104	3473106831504	23228952790176	107578640952064	355190527106576	845025856281432	1441253509696904	1722041314250626	1370413077902514	653076323891761	141104799067178			6662745366072879
13	6227020800	277238626560	5408825984640	61933677502464	467984563695504	2477852386714656	9496737357991984	26760513019710176	55579942348206192	84202283719181804	90593166598458256	65639759315609124	28740554887982161	5749087305575378		369775507472259699
14	87178291200	4251984710400	91369608491520	1159804136181888	9791136234916128	58476105017307792	255846596614760288	835717468040214672	2053182877358214784	3783144120884700868	5154932491086571992	5044068210067395188	3354403874646024602	1358434619275188921	252969604725106090	22162222616858076333

Technical details

Node id: n-9c1ae989b8f28a11 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-c19223d7fb5146f9

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:      1,        2;
  2:      2,        9,        10;
  3:      6,       44,       105,        82;
  4:     24,      250,       946,      1553,        938;
  5:    120,     1644,      8680,     22276,      27953,      13778;
  6:    720,    12348,     84680,    300172,     583866,     593561,      247210;
  7:   5040,   104544,    889308,   4063760,   10848010,   16999020,    14530169,     5240338;
  8:  40320,   986256,  10077048,  56786452,  194430720,  416285390,   546310786,   402923553,   128149802;
  9: 362880, 10265760, 123023736, 828862608, 3485617372, 9537606780, 17046350780, 19245766516, 12483933633, 3551246162;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n+(3*n+1)*x+(n+4)*x^2-(3*n-5)*x^3-(2*n-2)*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 >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (3*n + 1)*x + (n + 4)*x^2 - (3*n - 5)*x^3 - (2*n - 2)*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, 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] = ((n + (3*n + 1)*x + (n + 4)*x^2 - (3*n - 5)*x^3 - (2*n - 2)*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 - 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 [-1, 0)
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)

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