A200536 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 38\,x + 86\,x^{2} + 82\,x^{3} + 28\,x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	26	28													60
2	36	312	1012	1456	784											3600
3	216	2808	15192	43784	70896	61152	21952									216000
4	1296	22464	170208	736320	1989136	3436160	3706752	2283008	614656							12960000
5	7776	168480	1641600	9472320	35844960	92951456	167276480	206286080	166835200	79905280	17210368					777600000
6	46656	1213056	14447808	104232960	507314880	1754902656	4424055616	8189545728	11048190720	10593157120	6852185088	2684817408	481890304			46656000000
7	279936	8491392	119532672	1035006336	6158484864	26638125696	86376619392	213299994752	403090890496	580119181824	625884535808	490874116096	264557776896	87704035328	13492928512	2799360000000

All 15 rows

Technical details

Node id: n-61ae048ec6ed2266 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-be5ce6bf66b9c063

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        6,        26,         28;
  2:       36,       312,       1012,        1456,          784;
  3:      216,      2808,      15192,       43784,        70896,         61152,          21952;
  4:     1296,     22464,     170208,      736320,      1989136,       3436160,        3706752,        2283008,          614656;
  5:     7776,    168480,    1641600,     9472320,     35844960,      92951456,      167276480,      206286080,       166835200,        79905280,        17210368;
  6:    46656,   1213056,   14447808,   104232960,    507314880,    1754902656,     4424055616,     8189545728,     11048190720,     10593157120,      6852185088,       2684817408,        481890304;
  7:   279936,   8491392,  119532672,  1035006336,   6158484864,   26638125696,    86376619392,   213299994752,    403090890496,    580119181824,    625884535808,     490874116096,     264557776896,      87704035328,      13492928512;
  8:  1679616,  58226688,  945810432,  9555646464,  67207988736,  348929538048,  1383288560640,  4271459592192,  10382890549504,  19933478096896,  30124950876160,   35461579718656,   31874840682496,   21149201661952,    9768880242688,   2806529130496,    377801998336;
  9: 10077696, 393030144, 7235785728, 83555297280, 678177432576, 4108543036416, 19253723037696, 71364287195136, 212087372391936, 509156891449856, 989741071162368, 1554155587805184, 1956748741312512, 1948565299838976, 1500988063776768, 863007707627520, 348765216178176, 88405667610624, 10578455953408;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+38*x+86*x^2+82*x^3+28*x^4)*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[((6 + 38*x + 86*x^2 + 82*x^3 + 28*x^4)*P[n - 1, 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] = ((6 + 38*x + 86*x^2 + 82*x^3 + 28*x^4)*v[i - 1])/(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 basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    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.