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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	256	8704	132352	1185792	6935040	27675648	76349952	143824896	177106176	128770560	41990400							603979776
9	512	18944	316928	3165696	20984832	96961536	318753792	745749504	1217161728	1320178176	856604160	251942400						4831838208
10	1024	40960	747520	8232960	60963840	319832064	1219276800	3404021760	6908820480	9943326720	9634277376	5643509760	1511654400					38654705664
11	2048	88064	1740800	20951040	171325440	1005447168	4357545984	14123704320	34241771520	61339576320	78928515072	69092683776	36884367360	9069926400				309237645312
12	4096	188416	4009984	52346880	468357120	3038846976	14747774976	54392684544	153225768960	328129781760	525894488064	611756457984	488324837376	239446056960	54419558400			2473901162496
13	8192	401408	9150464	128753664	1250795520	8887836672	47728631808	197272018944	632807645184	1575614177280	3020567666688	4378879844352	4647188422656	3408841138176	1545515458560	326517350400		19791209299968
14	16384	851968	20709376	312410112	3274113024	25280446464	148784283648	680915828736	2449247404032	6948074225664	15494820397056	26881165688832	35567655911424	34700812812288	23544077746176	9926127452160	1959104102400	158329674399744

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.