A075505 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(10 + 11\,x - 9\,x^{2} - 10\,x^{3} - 10\,n x - 20\,n x^{2} - 10\,n x^{3}\right)\,P_{n-1}(x) + \left(10\,x + 30\,x^{2} + 30\,x^{3} + 10\,x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 100 + 230\,x + 131\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	100	230	131								461
1	1000	3700	4460	1761							10921
2	10000	55000	107500	90100	27601						290201
3	100000	810000	2330000	3136500	2023150	506651					8906301
4	1000000	12300000	49510000	95390000	97124000	50618210	10674211				316616421
5	10000000	197000000	1068600000	2755010000	3925090000	3186236600	1386243480	251686881			12779866961
6	100000000	3350000000	23675000000	78077000000	145579510000	163400686000	109920044200	40971070760	6524202561		571597513521
7	1000000000	60100000000	538100000000	2202005000000	5155782500000	7501565770000	6936643960000	3981587335000	1297723370450	183991725451	27858499660901

All 15 rows

Technical details

Node id: n-8857fecd171cfa47 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-5989189fb55e3737

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:          100,            230,             131;
  1:         1000,           3700,            4460,             1761;
  2:        10000,          55000,          107500,            90100,            27601;
  3:       100000,         810000,         2330000,          3136500,          2023150,            506651;
  4:      1000000,       12300000,        49510000,         95390000,         97124000,          50618210,          10674211;
  5:     10000000,      197000000,      1068600000,       2755010000,       3925090000,        3186236600,        1386243480,         251686881;
  6:    100000000,     3350000000,     23675000000,      78077000000,     145579510000,      163400686000,      109920044200,       40971070760,        6524202561;
  7:   1000000000,    60100000000,    538100000000,    2202005000000,    5155782500000,     7501565770000,     6936643960000,     3981587335000,     1297723370450,     183991725451;
  8:  10000000000,  1123000000000,  12507100000000,   62286300000000,  178284030000000,   323129428700000,   385308833370000,   302866758360000,   151572611975500,   43883018851550,    5605930566051;
  9: 100000000000, 21570000000000, 296726000000000, 1777534100000000, 6107330800000000, 13423168452000000, 19829000251600000, 20017443068270000, 13675386419355000, 6063954584615500, 1578554838246660, 183428104316161;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (10-(10*n-11)*x-(20*n+9)*x^2-(10*n+10)*x^3)*P(n-1,x) + (10*x+30*x^2+30*x^3+10*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 100 + 230*x + 131*x^2.

%t Mathematica

P[0, x_] := 100 + 230*x + 131*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((10 - (10*n - 11)*x - (20*n + 9)*x^2 - (10*n + 10)*x^3)*P[n - 1, x] + (10*x + 30*x^2 + 30*x^3 + 10*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 100 + 230*x + 131*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((10 - (10*n - 11)*x - (20*n + 9)*x^2 - (10*n + 10)*x^3)*v[i - 1] + (10*x + 30*x^2 + 30*x^3 + 10*x^4)*deriv(v[i - 1], x))/(1 + x)); 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 Stirling2 triangle with scaled diagonals (powers of 10).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[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.