A360282 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3\,n - 6\,n x - 3\,n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-2\,x^{2} - 4\,x^{3} - 2\,x^{4} + 5\,n x^{2} + 10\,n x^{3} + 5\,n x^{4} - 2\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-2}(x) + \left(6 + 22\,x + 26\,x^{2} + 10\,x^{3} - 16\,n - 56\,n x - 64\,n x^{2} - 24\,n x^{3} + 8\,n^{2} + 28\,n^{2} x + 32\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	3	5													8
3	0	10	32	25												67
4	0	35	165	255	129											584
5	0	126	784	1806	1824	681										5221
6	0	462	3570	10920	16520	12355	3653									47480
7	0	1716	15840	60390	121680	136620	81024	19825								437095
8	0	6435	69069	315315	793485	1188495	1059345	520191	108545							4060880
9	0	24310	297440	1581580	4772768	8938930	10638320	7855540	3290240	598417						37997545
10	0	92378	1268982	7701408	27099072	60918858	90720630	89489400	56378520	20582595	3317445					357569288
11	0	352716	5374720	36657270	147345120	386502480	691262208	853619130	718601520	394649190	127671840	18474633				3380510827
12	0	1352078	22632610	171361190	774589530	2322382920	4849028184	7194106920	7583535960	5565955395	2708751045	786653439	103274625			32083623896
13	0	5200300	94853472	789405540	3963442560	13369903470	31920720480	55305543216	70060193280	64398951540	41887248480	18299087730	4820802336	579168825		305494521229
14	0	20058300	395965700	3592664400	19836143184	74353666530	199798371630	395946888480	585883446720	647284701492	527322844620	307897040880	122012570160	29411279235	3256957317	2917012598648

Technical details

Node id: n-a22fbb70b22a479b · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-d8f957d0e068b855

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: 0,     1;
  2: 0,     3,      5;
  3: 0,    10,     32,      25;
  4: 0,    35,    165,     255,     129;
  5: 0,   126,    784,    1806,    1824,     681;
  6: 0,   462,   3570,   10920,   16520,   12355,     3653;
  7: 0,  1716,  15840,   60390,  121680,  136620,    81024,   19825;
  8: 0,  6435,  69069,  315315,  793485, 1188495,  1059345,  520191,  108545;
  9: 0, 24310, 297440, 1581580, 4772768, 8938930, 10638320, 7855540, 3290240, 598417;

%F Formula

The row polynomials satisfy (2*n^2-3*n+(4*n^2-6*n)*x+(2*n^2-3*n)*x^2)*P(n,x) = -((2*n^2-5*n+2)*x^2+(4*n^2-10*n+4)*x^3+(2*n^2-5*n+2)*x^4)*P(n-2,x) + (8*n^2-16*n+6+(28*n^2-56*n+22)*x+(32*n^2-64*n+26)*x^2+(12*n^2-24*n+10)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-((2*n^2 - 5*n + 2)*x^2 + (4*n^2 - 10*n + 4)*x^3 + (2*n^2 - 5*n + 2)*x^4)*P[n - 2, x] + (8*n^2 - 16*n + 6 + (28*n^2 - 56*n + 22)*x + (32*n^2 - 64*n + 26)*x^2 + (12*n^2 - 24*n + 10)*x^3)*P[n - 1, x])/(2*n^2 - 3*n + (4*n^2 - 6*n)*x + (2*n^2 - 3*n)*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; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (-((2*n^2 - 5*n + 2)*x^2 + (4*n^2 - 10*n + 4)*x^3 + (2*n^2 - 5*n + 2)*x^4)*v[i - 2] + (8*n^2 - 16*n + 6 + (28*n^2 - 56*n + 22)*x + (32*n^2 - 64*n + 26)*x^2 + (12*n^2 - 24*n + 10)*x^3)*v[i - 1])/(2*n^2 - 3*n + (4*n^2 - 6*n)*x + (2*n^2 - 3*n)*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 Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.