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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	256	7168	89856	664320	3208800	10584000	24150000	37650000	38390625	23125000	6250000							144120025
9	512	15616	215552	1777920	9739200	37212000	101220000	196050000	265031250	238203125	128125000	31250000						1008840175
10	1024	33792	509184	4633600	28368000	123120000	388500000	898200000	1510312500	1801562500	1447265625	703125000	156250000					7061881225
11	2048	72704	1187328	11813120	79904000	388080000	1392600000	3738900000	7511625000	11154687500	11902343750	8642578125	3828125000	781250000				49433168575
12	4096	155648	2738176	29562880	218873600	1175680000	4725600000	14440800000	33717750000	59867500000	79578125000	76796875000	50869140625	20703125000	3906250000			346032180025
13	8192	331776	6254592	72816640	585561600	3445728000	15329600000	52509600000	139639500000	288323750000	458493750000	551484375000	485722656250	295751953125	111328125000	19531250000		2422225260175
14	16384	704512	14168064	176906240	1535206400	9819264000	47887840000	181667200000	541827000000	1274845000000	2358606250000	3395437500000	3728867187500	3020117187500	1701416015625	595703125000	97656250000	16955576821225

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.