A161198 → basis A013609 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 + 2\,x + 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

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

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	7														10
3	15	62	63													140
4	105	599	1123	693												2520
5	945	6756	17894	20836	9009											55440
6	10395	88491	298174	497606	411639	135135										1441440
7	135135	1327098	5380609	11537836	13810609	8754618	2297295									43243200
8	2027025	22474035	105923997	275299111	426378019	393720841	200797167	43648605								1470268800
9	34459425	424626120	2272662684	6904488824	13029707654	15647192696	11681689500	4958766792	916620705							55870214400
10	654729075	8860463055	52946991756	183456529388	406367688378	596979937266	581837532508	362895427548	131467429611	21082276215						2346549004800
11	13749310575	202437951030	1333398403251	5176261911048	13120134690638	22695770892036	27143086614318	22166742291208	11833201710531	3729413540790	527056905375					107941254220800
12	316234143225	5027304259215	36133987952583	155055780841881	441522169482970	876246367164054	1237076806214286	1242698411284370	870665681204829	405272957622507	112816474424955	14230536445125				5397062711040000
13	7905853580625	134853396633900	1049141522331810	4924280171671932	15534671881488799	34710302094107480	56338064803114716	66942687662323544	57804895957367455	35381128695502716	14573327631676578	3627441169451820	412685556908625			291441386396160000
14	213458046676875	3886123170114675	32507276398609770	165478951827428274	572088826122027465	1418752984867054729	2597147114601429212	3553932575779291884	3635955508380951149	2747242249456964437	1490294835615851802	549714068157173058	123593186289539295	12793252264167375		16903600410977280000
15	6190283353629375	119741687473662450	1070953080173467605	5871629724149542356	22051381367843929527	60022767823171493486	122136114824054253205	188769899479446628632	222722484743764215493	199656557010823356590	133877544464939528679	65121437551881128148	21724766651583350469	4449579463215650610	422177324717523375	1048023225480591360000

Technical details

Node id: n-b727058f56ad81e9 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-393ad86efca4e577

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:         3,          7;
   3:        15,         62,          63;
   4:       105,        599,        1123,          693;
   5:       945,       6756,       17894,        20836,         9009;
   6:     10395,      88491,      298174,       497606,       411639,       135135;
   7:    135135,    1327098,     5380609,     11537836,     13810609,      8754618,      2297295;
   8:   2027025,   22474035,   105923997,    275299111,    426378019,    393720841,    200797167,     43648605;
   9:  34459425,  424626120,  2272662684,   6904488824,  13029707654,  15647192696,  11681689500,   4958766792,    916620705;
  10: 654729075, 8860463055, 52946991756, 183456529388, 406367688378, 596979937266, 581837532508, 362895427548, 131467429611, 21082276215;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(4*n+2)*x+(2*n+3)*x^2)*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[((2*n - 1 + (4*n + 2)*x + (2*n + 3)*x^2)*P[n - 1, 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] = ((2*n - 1 + (4*n + 2)*x + (2*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, 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 basis_map_A013609 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (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.