A156365 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 10\,x^{2} + 6\,x^{3} + x^{4} + n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} - 2\,x^{4} - x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 10\,x + 13\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	10	13															24
1	1	25	91	75														192
2	1	56	426	896	541													1920
3	1	119	1674	6734	9829	4683												23040
4	1	246	5979	40804	108039	120198	47293											322560
5	1	501	20157	218257	931347	1815951	1628911	545835										5160960
6	1	1012	65488	1077604	6940090	21082204	32355928	24287692	7087261									92897280
7	1	2035	207604	5037868	46947022	208222234	486156100	613463356	395661817	102247563								1857945600
8	1	4082	647241	22659000	296642754	1842924876	6136494618	11574468984	12380984925	6997344146	1622632573							40874803200
9	1	8177	1994799	99086871	1782958698	15072518058	68693342238	181584806286	286442939589	265649652437	133576402051	28091567595						980995276800
10	1	16368	6098886	424356440	10319962875	116189225088	704349359364	2504820349968	5463640455495	7393036903760	6047764375446	2738467744728	526858348381					25505877196800
11	1	32751	18542190	1789089098	58026149835	855754001613	6755843594868	31417127765820	90971165505159	168409077619865	199275523508286	145757189279706	60021703450765	10641342970443				714164561510400
12	1	65518	56150599	7453424444	318960138833	6080833298018	61505204538399	366276551657352	1368705438963627	3328132190876114	5341431828034781	5611130957968508	3710223743860099	1400840435357854	230283190977853			21424936845312000
13	1	131053	169565617	30769109693	1722182716901	41989309817153	537100199223429	4030528390093761	19037091965154819	59113250232744263	123546229987058291	174778573526593943	164987862267785047	99518962614537163	34688982753471511	5315654681981355		685597979049984000
14	1	262124	511055820	126130454620	9166961912000	283304377123452	4534452981158668	42331923804943340	248717240501091930	966044384820979460	2556306677803121668	4675854523482642132	5908587682895188040	5062161291530026900	2806639655167096740	908490089703263204	130370767029135901	23310331287699456000

Technical details

Node id: n-4cbdf6d8927688ef · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-31922ed48acb43c6

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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,   10,      13;
  1: 1,   25,      91,       75;
  2: 1,   56,     426,      896,        541;
  3: 1,  119,    1674,     6734,       9829,        4683;
  4: 1,  246,    5979,    40804,     108039,      120198,       47293;
  5: 1,  501,   20157,   218257,     931347,     1815951,     1628911,       545835;
  6: 1, 1012,   65488,  1077604,    6940090,    21082204,    32355928,     24287692,      7087261;
  7: 1, 2035,  207604,  5037868,   46947022,   208222234,   486156100,    613463356,    395661817,    102247563;
  8: 1, 4082,  647241, 22659000,  296642754,  1842924876,  6136494618,  11574468984,  12380984925,   6997344146,   1622632573;
  9: 1, 8177, 1994799, 99086871, 1782958698, 15072518058, 68693342238, 181584806286, 286442939589, 265649652437, 133576402051, 28091567595;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(n+6)*x+(3*n+10)*x^2+(3*n+6)*x^3+(n+1)*x^4)*P(n-1,x) + (x+2*x^2-2*x^4-x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 10*x + 13*x^2.

%t Mathematica

P[0, x_] := 1 + 10*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (n + 6)*x + (3*n + 10)*x^2 + (3*n + 6)*x^3 + (n + 1)*x^4)*P[n - 1, x] + (x + 2*x^2 - 2*x^4 - 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 + 10*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 6)*x + (3*n + 10)*x^2 + (3*n + 6)*x^3 + (n + 1)*x^4)*v[i - 1] + (x + 2*x^2 - 2*x^4 - 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 T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A142075 → x ↦ x/(x+1)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003