A094262 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 9\,x + 33\,x^{2} + 63\,x^{3} + 66\,x^{4} + 36\,x^{5} + 8\,x^{6} - 2\,n x - 14\,n x^{2} - 38\,n x^{3} - 50\,n x^{4} - 32\,n x^{5} - 8\,n x^{6}\right)\,P_{n-1}(x) + \left(x + 8\,x^{2} + 26\,x^{3} + 44\,x^{4} + 41\,x^{5} + 20\,x^{6} + 4\,x^{7}\right)\,P_{n-1}'(x)\)

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

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	4	4													9
2	1	10	36	56	32											135
3	1	20	146	528	1024	1024	416									3159
4	1	38	478	3016	11000	24352	32416	23936	7552							102789
5	1	72	1444	14140	81376	298576	721984	1150528	1166080	682752	176128					4293081
6	1	138	4224	60280	502668	2714832	10005408	25773120	46552320	57933056	47408640	23000064	5018624			218973375
7	1	268	12214	244552	2810956	20946448	107987904	399359232	1078851456	2138152448	3080602112	3144452096	2157759488	893702144	168968192	13193849511

All 15 rows

Technical details

Node id: n-56cb29b4e81fea25 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-3611becd7eab4c6b

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,    4,      4;
  2: 1,   10,     36,      56,       32;
  3: 1,   20,    146,     528,     1024,      1024,        416;
  4: 1,   38,    478,    3016,    11000,     24352,      32416,       23936,         7552;
  5: 1,   72,   1444,   14140,    81376,    298576,     721984,     1150528,      1166080,       682752,        176128;
  6: 1,  138,   4224,   60280,   502668,   2714832,   10005408,    25773120,     46552320,     57933056,      47408640,      23000064,        5018624;
  7: 1,  268,  12214,  244552,  2810956,  20946448,  107987904,   399359232,   1078851456,   2138152448,    3080602112,    3144452096,     2157759488,      893702144,      168968192;
  8: 1,  526,  35250,  966360, 14823900, 146354520, 1003132640,  4990836320,  18507620160,  51888892160,  110486117888,  177828123648,   213045027840,   184310446080,   108896133120,    39348527104,    6563282944;
  9: 1, 1040, 102008, 3767524, 75436020, 959955360, 8459217320, 54344440480, 262880626240, 977013635200, 2822041111168, 6362832824320, 11173028056064, 15134028214272, 15514954168320, 11646491336704, 6041406586880, 1936045801472, 288909131776;

%F Formula

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-9)*x-(14*n-33)*x^2-(38*n-63)*x^3-(50*n-66)*x^4-(32*n-36)*x^5-(8*n-8)*x^6)*P(n-1,x) + (x+8*x^2+26*x^3+44*x^4+41*x^5+20*x^6+4*x^7)*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[((1 - (2*n - 9)*x - (14*n - 33)*x^2 - (38*n - 63)*x^3 - (50*n - 66)*x^4 - (32*n - 36)*x^5 - (8*n - 8)*x^6)*P[n - 1, x] + (x + 8*x^2 + 26*x^3 + 44*x^4 + 41*x^5 + 20*x^6 + 4*x^7)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
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] = ((1 - (2*n - 9)*x - (14*n - 33)*x^2 - (38*n - 63)*x^3 - (50*n - 66)*x^4 - (32*n - 36)*x^5 - (8*n - 8)*x^6)*v[i - 1] + (x + 8*x^2 + 26*x^3 + 44*x^4 + 41*x^5 + 20*x^6 + 4*x^7)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.