A109692 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-3\,x - 3\,x^{2} + 2\,n + 10\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	5														7
2	8	46	65													119
3	48	444	1356	1365												3213
4	384	4944	23724	50244	39585											118881
5	3840	63648	420168	1380228	2254878	1464645										5587407
6	46080	936576	7906176	35470296	89168796	119045250	65909025									318482199
7	645120	15554304	160324992	915611472	3128288832	6392579688	7232124600	3493178325								21338307333
8	10321920	288221184	3514012416	24429608064	105904921104	293106893760	505661354568	497050453800	213083877825							1643049664641
9	185794560	5900193792	83139485184	682199801856	3591931536288	12583363643856	29326280051664	43837541633592	38131991113050	14702787569925						142945320823767
10	3715891200	132310056960	2117104625664	20045736396288	124368015468672	528246001171296	1555444601610192	3134874396649968	4138130528047584	3230219067103350	1132114642884225					13865696119905399
11	81749606400	3226672005120	57822656606208	620960093899776	4439983933995264	22192693340605632	79120691334984384	201180027863165616	357503195332294128	422805914360318340	299475142847237700	96229744645159125				1483629484829877693
12	1961990553600	85042841518080	1687824255025152	20280549317971968	164308903148565504	945543146036094720	3962817072715948992	12186544962869522496	27289819279249461360	43395139110550994064	46508353463843310420	30160702156276925100	8949366251999798625			173584649725095690081
13	51011754393600	2409274925383680	52472757623980032	697764532024811520	6320366962977871872	41179321014943578624	198533101640260240512	717094693378918433088	1940376342510307767456	3884545364078521443024	5592126240225576471384	5481521955911374405020	3278914440335964199350	903885991451979661125		22045250515087152640287
14	1428329123020800	73019979139645440	1731848180338262016	25256937477708546048	253026608954084868096	1841940987383008235520	10047472836556136804352	41718759493398082342272	132493859168590726695360	320268291527822147057376	579994979410874978488368	763024374950106318721416	689295497523746807728980	382710481757275528240650	98523573068265783062625	3020199320566939911719319

Technical details

Node id: n-f0b30e5b06c6098c · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-88408b4d23225179

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 0)
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;
  1:         2,          5;
  2:         8,         46,          65;
  3:        48,        444,        1356,         1365;
  4:       384,       4944,       23724,        50244,         39585;
  5:      3840,      63648,      420168,      1380228,       2254878,        1464645;
  6:     46080,     936576,     7906176,     35470296,      89168796,      119045250,       65909025;
  7:    645120,   15554304,   160324992,    915611472,    3128288832,     6392579688,     7232124600,     3493178325;
  8:  10321920,  288221184,  3514012416,  24429608064,  105904921104,   293106893760,   505661354568,   497050453800,   213083877825;
  9: 185794560, 5900193792, 83139485184, 682199801856, 3591931536288, 12583363643856, 29326280051664, 43837541633592, 38131991113050, 14702787569925;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2*n+(10*n-3)*x+(8*n-3)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (10*n - 3)*x + (8*n - 3)*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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + (10*n - 3)*x + (8*n - 3)*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 Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    
    ↑ Liu–Wang theorem (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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.