A225469 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 15\,x + 20\,x^{2} + 8\,x^{3} - 4\,n x - 8\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	4														7
3	9	28	20													57
4	27	160	260	128												575
5	81	904	2536	2720	1008											7249
6	243	5356	22984	40096	31504	9280										109463
7	729	33604	205116	515744	643536	396736	96704									1892169
8	2187	220648	1835532	6243616	11008304	10658432	5397440	1120768								36486927
9	6561	1492048	16599248	73674432	172447136	233312512	184166144	79014912	14274816							774987809
10	19683	10260628	152540816	864148160	2583415904	4582487680	5013081856	3333035008	1238706944	197833728						17975530407
11	59049	71194540	1430985316	10193229248	37900574048	84657660800	120344793472	110213592064	63190600960	20689857536	2958521344					451651068377
12	177147	496236016	13745029396	121715728960	552000625824	1510113483264	2678326289792	3167605659648	2490426179328	1253678166016	366526481408	47415508992				12202049565791
13	531441	3466566232	135307976376	1476005788256	8045739169040	26421701781760	56858792301312	83379662169088	84280081035008	57981986224128	25993225861120	6860162850816	809838505984			352245970760561
14	1594323	24242580220	1363565846616	18198934571104	117936609391600	458014062574784	1172036931599104	2072397046557696	2585747767022848	2279305840059392	1392723029743616	562399950299136	135203122352128	14670950907904		10810022055100471
15	4782969	169621534036	14029438199180	228186267624288	1744024251482800	7919380306110272	23735166714479936	49587858440895488	74263314133486336	80527362917297152	62875475099943936	34531041174904832	12678680895000576	2797452570443776	280760761434112	351182902597619689

Technical details

Node id: n-29fb7d469859e60a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-5087f04152d0b6ee

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,        4;
   3:     9,       28,        20;
   4:    27,      160,       260,       128;
   5:    81,      904,      2536,      2720,       1008;
   6:   243,     5356,     22984,     40096,      31504,       9280;
   7:   729,    33604,    205116,    515744,     643536,     396736,      96704;
   8:  2187,   220648,   1835532,   6243616,   11008304,   10658432,    5397440,    1120768;
   9:  6561,  1492048,  16599248,  73674432,  172447136,  233312512,  184166144,   79014912,   14274816;
  10: 19683, 10260628, 152540816, 864148160, 2583415904, 4582487680, 5013081856, 3333035008, 1238706944, 197833728;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (3-(4*n-15)*x-(8*n-20)*x^2-(4*n-8)*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3 - (4*n - 15)*x - (8*n - 20)*x^2 - (4*n - 8)*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3 - (4*n - 15)*x - (8*n - 20)*x^2 - (4*n - 8)*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.