A124848 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(5\,x + 12\,x^{2} + 9\,x^{3} + 2\,x^{4} + n + 2\,n x - n x^{2} - 4\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	5														6
2	1	10	19													30
3	1	15	57	63												136
4	1	20	114	252	192											579
5	1	25	190	630	960	552										2358
6	1	30	285	1260	2880	3312	1520									9288
7	1	35	399	2205	6720	11592	10640	4048								35640
8	1	40	532	3528	13440	30912	42560	32384	10496							133893
9	1	45	684	5292	24192	69552	127680	145728	94464	26624						494262
10	1	50	855	7560	40320	139104	319200	485760	472320	266240	66304					1797714
11	1	55	1045	10395	63360	255024	702240	1335840	1731840	1464320	729344	162560				6456024
12	1	60	1254	13860	95040	437184	1404480	3206016	5195520	5857280	4376064	1950720	393216			22930695
13	1	65	1482	18018	137280	710424	2608320	6946368	13508352	19036160	18962944	12679680	5111808	940032		80660934
14	1	70	1729	22932	192192	1105104	4564560	13892736	31519488	53301248	66370304	59171840	35782656	13160448	2224128	281309436

Technical details

Node id: n-a667d51220d86c73 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-541a6ed819451226

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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: 1,  5;
  2: 1, 10,  19;
  3: 1, 15,  57,   63;
  4: 1, 20, 114,  252,   192;
  5: 1, 25, 190,  630,   960,   552;
  6: 1, 30, 285, 1260,  2880,  3312,   1520;
  7: 1, 35, 399, 2205,  6720, 11592,  10640,   4048;
  8: 1, 40, 532, 3528, 13440, 30912,  42560,  32384, 10496;
  9: 1, 45, 684, 5292, 24192, 69552, 127680, 145728, 94464, 26624;

%F Formula

The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n+(2*n+5)*x-(n-12)*x^2-(4*n-9)*x^3-(2*n-2)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*x^5)*d/dx 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[((n + (2*n + 5)*x - (n - 12)*x^2 - (4*n - 9)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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] = ((n + (2*n + 5)*x - (n - 12)*x^2 - (4*n - 9)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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 read by rows: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).?
  - 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.