A286785 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(56 + 112\,x + 56\,x^{2} + 30\,n + 60\,n x + 30\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(154 + 518\,x + 574\,x^{2} + 210\,x^{3} + 93\,n + 309\,n x + 339\,n x^{2} + 123\,n x^{3} + 14\,n^{2} + 46\,n^{2} x + 50\,n^{2} x^{2} + 18\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(2 + 36\,x + 96\,x^{2} + 92\,x^{3} + 30\,x^{4} + n + 12\,n x + 30\,n x^{2} + 28\,n x^{3} + 9\,n x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 14 + 42\,x + 30\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	14	42	30								86
1	42	198	297	143							680
2	132	858	2002	2002	728						5722
3	429	3575	11440	17680	13260	3876					50260
4	1430	14586	59670	125970	145350	87210	21318				455534
5	4862	58786	293930	791350	1243550	1144066	572033	120175			4228752
6	16796	235144	1392776	4576264	9152528	11440660	8748740	3749460	690690		40003058
7	58786	936054	6418656	24961440	60843510	96633810	100212840	65552760	24582285	4032015	384232156

All 15 rows

Technical details

Node id: n-8f81b8435f04f925 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-caa98baf3ef4c119

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:     14,       42,        30;
  1:     42,      198,       297,       143;
  2:    132,      858,      2002,      2002,        728;
  3:    429,     3575,     11440,     17680,      13260,       3876;
  4:   1430,    14586,     59670,    125970,     145350,      87210,      21318;
  5:   4862,    58786,    293930,    791350,    1243550,    1144066,     572033,     120175;
  6:  16796,   235144,   1392776,   4576264,    9152528,   11440660,    8748740,    3749460,     690690;
  7:  58786,   936054,   6418656,  24961440,   60843510,   96633810,  100212840,   65552760,   24582285,    4032015;
  8: 208012,  3714500,  28973100, 130378950,  375798150,  726543090,  955977750,  846723150,  483841800,  161280600,   23841480;
  9: 742900, 14709420, 128707425, 658679175, 2195597250, 5015206350, 8024330160, 9006901200, 6959878200, 3530373000, 1059111900, 142498692;

%F Formula

The row polynomials satisfy (4*n^2+30*n+56+(8*n^2+60*n+112)*x+(4*n^2+30*n+56)*x^2)*P(n,x) = (14*n^2+93*n+154+(46*n^2+309*n+518)*x+(50*n^2+339*n+574)*x^2+(18*n^2+123*n+210)*x^3)*P(n-1,x) + (n+2+(12*n+36)*x+(30*n+96)*x^2+(28*n+92)*x^3+(9*n+30)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 42*x + 30*x^2.

%t Mathematica

P[0, x_] := 14 + 42*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 + 93*n + 154 + (46*n^2 + 309*n + 518)*x + (50*n^2 + 339*n + 574)*x^2 + (18*n^2 + 123*n + 210)*x^3)*P[n - 1, x] + (n + 2 + (12*n + 36)*x + (30*n + 96)*x^2 + (28*n + 92)*x^3 + (9*n + 30)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56 + (8*n^2 + 60*n + 112)*x + (4*n^2 + 30*n + 56)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 14 + 42*x + 30*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 + 93*n + 154 + (46*n^2 + 309*n + 518)*x + (50*n^2 + 339*n + 574)*x^2 + (18*n^2 + 123*n + 210)*x^3)*v[i - 1] + (n + 2 + (12*n + 36)*x + (30*n + 96)*x^2 + (28*n + 92)*x^3 + (9*n + 30)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56 + (8*n^2 + 60*n + 112)*x + (4*n^2 + 30*n + 56)*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 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[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.