A272866 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 35\,x + 84\,x^{2} + 83\,x^{3} + 29\,x^{4}\right)\,P_{n-1}(x)\)

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

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	25	29													59
2	25	250	915	1450	841											3481
3	125	1875	11550	37375	66990	63075	24389									205379
4	625	12500	108250	530000	1604275	3074000	3641530	2438900	707281							12117361
5	3125	78125	871875	5718750	24410625	70846875	141581625	192378750	170113275	88410125	20511149					714924299
6	15625	468750	6403125	52656250	290306250	1130343750	3186988125	6555993750	9765902250	10273866250	7246093845	3076672350	594823321			42180533641
7	78125	2734375	44187500	436953125	2953628125	14436406250	52612415625	145234640625	305152010625	485640706250	576288290725	494477829125	290027646860	104094081175	17249876309	2488651484819

All 15 rows

Technical details

Node id: n-89fa4fd1d325b86a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8c53afe4f9846d4c

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,       25,         29;
  2:      25,      250,        915,        1450,          841;
  3:     125,     1875,      11550,       37375,        66990,         63075,         24389;
  4:     625,    12500,     108250,      530000,      1604275,       3074000,       3641530,        2438900,          707281;
  5:    3125,    78125,     871875,     5718750,     24410625,      70846875,     141581625,      192378750,       170113275,        88410125,        20511149;
  6:   15625,   468750,    6403125,    52656250,    290306250,    1130343750,    3186988125,     6555993750,      9765902250,     10273866250,      7246093845,       3076672350,        594823321;
  7:   78125,  2734375,   44187500,   436953125,   2953628125,   14436406250,   52612415625,   145234640625,    305152010625,    485640706250,    576288290725,     494477829125,     290027646860,     104094081175,      17249876309;
  8:  390625, 15625000,  291562500,  3368750000,  26973406250,  158694375000,  709627450000,  2460139375000,   6682386121875,  14268808375000,  23871867418000,   30963176895000,   30524444393450,   22111018622000,   11099403169860,   3449975261800,    500246412961;
  9: 1953125, 87890625, 1859765625, 24585937500, 227541093750, 1565500781250, 8297725406250, 34643520000000, 115494611034375, 309747736796875, 669868743999375, 1165408012800000, 1618985799464250, 1771598332901250, 1493481368809350, 935954495593500, 410633305535745, 112555442916225, 14507145975869;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+35*x+84*x^2+83*x^3+29*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 + 35*x + 84*x^2 + 83*x^3 + 29*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 + 35*x + 84*x^2 + 83*x^3 + 29*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_A013610 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.