A272866 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 40\,x + 106\,x^{2} + 112\,x^{3} + 41\,x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	30	41													76
2	25	300	1310	2460	1681											5776
3	125	2250	16575	63900	135915	151290	68921									438976
4	625	15000	155500	909000	3276150	7453800	10455820	8270520	2825761							33362176
5	3125	93750	1253125	9825000	50026250	172822500	410215250	660633000	690933025	423864150	115856201					2535525376
6	15625	562500	9206250	90562500	596259375	2767725000	9286827500	22695345000	40092480375	49933264500	41623459530	20854116180	4750104241			192699928576
7	78125	3281250	63546875	752062500	6075628125	35439468750	153912521875	505558275000	1262082679375	2382949878750	3349906928025	3400238211300	2355935847335	997521890610	194754273881	14645194571776

All 15 rows

Technical details

Node id: n-65ae6116fb370fbc · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-158304e21352a080

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:       5,        30,         41;
  2:      25,       300,       1310,        2460,         1681;
  3:     125,      2250,      16575,       63900,       135915,        151290,          68921;
  4:     625,     15000,     155500,      909000,      3276150,       7453800,       10455820,         8270520,         2825761;
  5:    3125,     93750,    1253125,     9825000,     50026250,     172822500,      410215250,       660633000,       690933025,        423864150,        115856201;
  6:   15625,    562500,    9206250,    90562500,    596259375,    2767725000,     9286827500,     22695345000,     40092480375,      49933264500,      41623459530,      20854116180,        4750104241;
  7:   78125,   3281250,   63546875,   752062500,   6075628125,   35439468750,   153912521875,    505558275000,   1262082679375,    2382949878750,    3349906928025,    3400238211300,     2355935847335,      997521890610,      194754273881;
  8:  390625,  18750000,  419375000,  5801250000,  55545437500,  390300750000,  2081847425000,   8598185250000,  27787575043750,   70505119050000,  139983420857000,  215199343926000,   251133009624700,   215075451536400,   127492797828440,    46741025731440,    7984925229121;
  9: 1953125, 105468750, 2675390625, 42356250000, 468959062500, 3855718125000, 24395622562500, 121448679750000, 482239177143750, 1538678441812500, 3954361252578750, 8166209226390000, 13450965621040500, 17432540647389000, 17386180929846900, 12876582576502800, 6669360109054845, 2155929811862670, 327381934393961;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+40*x+106*x^2+112*x^3+41*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[((5 + 40*x + 106*x^2 + 112*x^3 + 41*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] = ((5 + 40*x + 106*x^2 + 112*x^3 + 41*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_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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.