A358735 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2)coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 - 3\,x - 3\,x^{2} - x^{3}\right)\,P_{n-2}(x) + \left(2 + 4\,x + 2\,x^{2} + n + 3\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	7	22	17													46
3	33	159	253	133												578
4	191	1248	3040	3272	1313											9064
5	1304	10803	35652	58586	47936	15623										169904
6	10241	103050	430731	957216	1192795	790190	217409									3701632
7	90865	1077895	5466817	15365947	25849851	26026349	14520539	3462921								91861184
8	898409	12290988	73423208	250140448	531547598	721427812	610689088	294773904	62115169							2557306624
9	9791634	151909423	1045761576	4192631612	10787783976	18473565894	21053713120	15397764356	6557298558	1238840459						78909060608
10	116601199	2024241218	15792242625	72909056164	220586051474	456979374744	656477914798	645717281020	416176046463	158707649542	27192374929					2672678834176
11	1506023953	28942071919	252521571301	1320395488945	4597217208322	11190555341590	19432929257346	24073815851922	20849096808109	12021673357227	4153449483609	651378157837				98573480622080
12	20967734143	442088186028	4267838541848	24944579916424	98309127222651	275224361664776	561224425768424	839873930132544	915442478940197	708743166988524	369951016337952	116895888384152	16908639728833			3932248509546496
13	313009988192	7186465226599	76084135597412	491813797214638	2165594616242124	6859315601958689	16078237255991240	28238262643795828	37159677570044888	36185743232328009	25345028027741780	12091086832742654	3521258264889468	472790534249487		168692391988011008
14	4987192076929	123889719617030	1427767143708515	10117622478170420	49251461650715777	174220100922576242	461821281878299883	931921772837136280	1438546305669619339	1690382716763593322	1488389865584453553	952242689068087284	418452085787676019	113051683326834014	14166807387755777	7744121037410320384

Technical details

Node id: n-3ef7de8b16847400 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-39879bf7c0fc9429

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:       2,         3;
  2:       7,        22,         17;
  3:      33,       159,        253,        133;
  4:     191,      1248,       3040,       3272,        1313;
  5:    1304,     10803,      35652,      58586,       47936,       15623;
  6:   10241,    103050,     430731,     957216,     1192795,      790190,      217409;
  7:   90865,   1077895,    5466817,   15365947,    25849851,    26026349,    14520539,     3462921;
  8:  898409,  12290988,   73423208,  250140448,   531547598,   721427812,   610689088,   294773904,   62115169;
  9: 9791634, 151909423, 1045761576, 4192631612, 10787783976, 18473565894, 21053713120, 15397764356, 6557298558, 1238840459;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = -(1+3*x+3*x^2+x^3)*P(n-2,x) + (n+2+(3*n+4)*x+(2*n+2)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + 3*x + 3*x^2 + x^3)*P[n - 2, x] + (n + 2 + (3*n + 4)*x + (2*n + 2)*x^2)*P[n - 1, 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] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(1 + 3*x + 3*x^2 + x^3)*v[i - 2] + (n + 2 + (3*n + 4)*x + (2*n + 2)*x^2)*v[i - 1])/(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, -1/2)
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_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023