A142963 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 2\,x - 2\,x^{2} - 6\,x^{3} - 3\,x^{4} + 3\,n x + 9\,n x^{2} + 9\,n x^{3} + 3\,n x^{4}\right)\,P_{n-1}(x) + \left(x - 6\,x^{3} - 8\,x^{4} - 3\,x^{5}\right)\,P_{n-1}'(x)\)

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

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	3														4
2	1	12	15													28
3	1	33	135	111												280
4	1	78	744	1722	1071											3616
5	1	171	3258	15774	25317	12663										57184
6	1	360	12525	110760	345195	423792	177255									1069888
7	1	741	44433	661365	3537675	8027127	7973667	2866671								23111680
8	1	1506	149622	3547458	30259440	112701942	200034522	166780134	52587711							566062336
9	1	3039	486540	17662764	228487014	1304352882	3679287948	5349076380	3841638273	1078805223						15499800064
10	1	6108	1544739	83374032	1576558746	13170761352	55269378486	124672028688	153273558069	96665446332	24470577495					469183234048
11	1	12249	4823595	378458475	10170966330	120189391434	718191217926	2355594597990	4411463109525	4694353208685	2638799395839	608113846431				15557259028480
12	1	24534	14887284	1668556410	62338822965	1015006409244	8365852995696	38244755059956	102299117410395	163439929453230	153228319096284	77684275742706	16429992037551			560771270496256
13	1	49107	45569646	7196090202	367203094875	8067592990449	89492288589108	553151205957564	2027396287114983	4560950320870725	6345339437501694	5314508736820938	2453592498691869	479502707791383		21832375521132544
14	1	98256	138673257	30517775208	2096578519965	61106345874504	894523479379749	7303600537104528	35614138411217907	108278528380932960	209672485820140779	258102976317337224	195294932156928303	82764755306458632	15032820820737735	913021994811179008

Technical details

Node id: n-24b3ff6153ebbb7f · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-1f7a7250caf8ceb0

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,    3;
  2: 1,   12,     15;
  3: 1,   33,    135,      111;
  4: 1,   78,    744,     1722,      1071;
  5: 1,  171,   3258,    15774,     25317,      12663;
  6: 1,  360,  12525,   110760,    345195,     423792,     177255;
  7: 1,  741,  44433,   661365,   3537675,    8027127,    7973667,    2866671;
  8: 1, 1506, 149622,  3547458,  30259440,  112701942,  200034522,  166780134,   52587711;
  9: 1, 3039, 486540, 17662764, 228487014, 1304352882, 3679287948, 5349076380, 3841638273, 1078805223;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(3*n+2)*x+(9*n-2)*x^2+(9*n-6)*x^3+(3*n-3)*x^4)*P(n-1,x) + (x-6*x^3-8*x^4-3*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[((1 + (3*n + 2)*x + (9*n - 2)*x^2 + (9*n - 6)*x^3 + (3*n - 3)*x^4)*P[n - 1, x] + (x - 6*x^3 - 8*x^4 - 3*x^5)*D[P[n - 1, x], 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] = ((1 + (3*n + 2)*x + (9*n - 2)*x^2 + (9*n - 6)*x^3 + (3*n - 3)*x^4)*v[i - 1] + (x - 6*x^3 - 8*x^4 - 3*x^5)*deriv(v[i - 1], x))/(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 Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[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