A159854 → basis A013611 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 8\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 10\,x + 25\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	10	25								36
1	2	26	110	150							288
2	4	64	376	960	900						2304
3	8	152	1136	4176	7560	5400					18432
4	16	352	3184	15168	40176	56160	32400				147456
5	32	800	8480	49440	171360	353376	401760	194400			1179648
6	64	1792	21760	149760	639360	1734912	2923776	2799360	1166400		9437184
7	128	3968	54272	430080	2177280	7305984	16257024	23141376	19128960	6998400	75497472

All 15 rows

Technical details

Node id: n-7b844c4754d63ea9 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-fc9aafad96b08502

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,    10,     25;
  1:   2,    26,    110,     150;
  2:   4,    64,    376,     960,      900;
  3:   8,   152,   1136,    4176,     7560,     5400;
  4:  16,   352,   3184,   15168,    40176,    56160,     32400;
  5:  32,   800,   8480,   49440,   171360,   353376,    401760,    194400;
  6:  64,  1792,  21760,  149760,   639360,  1734912,   2923776,   2799360,    1166400;
  7: 128,  3968,  54272,  430080,  2177280,  7305984,  16257024,  23141376,   19128960,    6998400;
  8: 256,  8704, 132352, 1185792,  6935040, 27675648,  76349952, 143824896,  177106176,  128770560,  41990400;
  9: 512, 18944, 316928, 3165696, 20984832, 96961536, 318753792, 745749504, 1217161728, 1320178176, 856604160, 251942400;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2+8*x+6*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 10*x + 25*x^2.

%t Mathematica

P[0, x_] := 1 + 10*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 8*x + 6*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 + 10*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 8*x + 6*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_A013611 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.