A161198 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 + 4\,x + 5\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

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

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	9														12
3	15	78	99													192
4	105	741	1707	1287												3840
5	945	8244	26478	37188	19305											92160
6	10395	106749	431406	859194	844551	328185										2580480
7	135135	1585242	7636509	19366236	27303849	20312874	6235515									82575360
8	2027025	26616465	147837717	450860229	816248691	878073939	520103079	130945815								2972712960
9	34459425	499101480	3125419884	11064891384	24246013014	33700976856	29037452940	14188449672	3011753745							118908518400
10	654729075	10344413745	71860514796	288368433396	737296531866	1246468885614	1394236027260	995516867268	411934562955	75293843625						5231974809600
11	13749310575	234910373670	1788369981831	7995971000808	23269174870878	46082852958276	62933616484038	58550226948648	35529581238291	12703403915910	2032933777875					251134790860800
12	316234143225	5801668601085	47944910418543	235770062491683	767074181053626	1734711689295810	2783875914922878	3171730097856006	2515136949991485	1322536145976369	415156190452515	58955079558375				13059009124761600
13	7905853580625	154844973467100	1378474487097210	7380543785266908	26485726463582823	67147091845057656	123372960241242060	165593405809009368	161202056783323311	111032649099145260	51377525286580314	14343718892987340	1827607466309625			731304510986649600
14	213458046676875	4441707451772325	42328695276038970	244764340276414446	958672559430544185	2687000453114789871	5546923957400438268	8542329644804240916	9817037924847038541	8317549399526591283	5051270603009462058	2082738744567808542	522688125058942095	60311046388217625		43878270659198976000
15	6190283353629375	136280547735088050	1382991923817161505	8579670202677382884	36368256133160286975	111476552720397952734	254905810623630355257	441869898208338325944	583675637388712549749	584805259955917496142	437601076470705094587	237193894697797619748	88053811686582619725	20043104722321284450	2110886623587616875	2808209322188734464000

Technical details

Node id: n-6069d893b32d8be9 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-4b7e8d702e00ffa0

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,           9;
   3:        15,          78,          99;
   4:       105,         741,        1707,         1287;
   5:       945,        8244,       26478,        37188,        19305;
   6:     10395,      106749,      431406,       859194,       844551,        328185;
   7:    135135,     1585242,     7636509,     19366236,     27303849,      20312874,       6235515;
   8:   2027025,    26616465,   147837717,    450860229,    816248691,     878073939,     520103079,    130945815;
   9:  34459425,   499101480,  3125419884,  11064891384,  24246013014,   33700976856,   29037452940,  14188449672,   3011753745;
  10: 654729075, 10344413745, 71860514796, 288368433396, 737296531866, 1246468885614, 1394236027260, 995516867268, 411934562955, 75293843625;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(4*n+4)*x+(2*n+5)*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 + 4)*x + (2*n + 5)*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 + 4)*x + (2*n + 5)*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_A013610 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.