A159854 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 7\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 8\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	8	16								25
1	2	21	72	80							175
2	4	52	249	520	400						1225
3	8	124	758	2285	3400	2000					8575
4	16	288	2136	8360	18225	21000	10000				60025
5	32	656	5712	27400	78250	133125	125000	50000			420175
6	64	1472	14704	83360	293500	657500	915625	725000	250000		2941225
7	128	3264	36768	240240	1003800	2782500	5118750	6028125	4125000	1250000	20588575

All 15 rows

Technical details

Node id: n-96ed33e0df829224 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-6aa3fd3b580cc302

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1,     8,     16;
  1:   2,    21,     72,      80;
  2:   4,    52,    249,     520,     400;
  3:   8,   124,    758,    2285,    3400,     2000;
  4:  16,   288,   2136,    8360,   18225,    21000,     10000;
  5:  32,   656,   5712,   27400,   78250,   133125,    125000,     50000;
  6:  64,  1472,  14704,   83360,  293500,   657500,    915625,    725000,    250000;
  7: 128,  3264,  36768,  240240, 1003800,  2782500,   5118750,   6028125,   4125000,   1250000;
  8: 256,  7168,  89856,  664320, 3208800, 10584000,  24150000,  37650000,  38390625,  23125000,   6250000;
  9: 512, 15616, 215552, 1777920, 9739200, 37212000, 101220000, 196050000, 265031250, 238203125, 128125000, 31250000;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2+7*x+5*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 16*x^2.

%t Mathematica

P[0, x_] := 1 + 8*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 7*x + 5*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 + 8*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 7*x + 5*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, 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 basis_map_A013610 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    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

[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.