A008955 → basis A007318 → 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

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

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

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	3														5
2	10	33	27													70
3	100	520	897	513												2030
4	1700	12140	32409	38322	16929											101500
5	44200	402340	1461774	2649231	2394576	863379										7815500
6	1635400	18113180	83456458	204731049	281993175	206749071	63026667									859705000
7	81770000	1067563600	5966027720	18498741792	34368032601	38254777875	23619491379	6239640033								128096045000
8	5315050000	79939964000	525507506200	1972035792360	4620259810233	6920036767404	6470133285510	3452490990036	804913564257							24850632730000
9	435834100000	7421430198000	56121829640400	247364658484120	700303138593786	1320545363995107	1658516922498672	1337735986721082	628758943644942	131200910973891						6088405018850000
10	44019244100000	837167104098000	7160012263478400	36264318264616520	120450913353280506	274136012620856793	432939827335382379	468473236081062354	332389586639076624	139631839680996333	26371383105752091					1838698315692700000
11	5370347780200000	112831063016256000	1076953102440178800	6164129808308466640	23507240767402036092	62714165484591691704	119433710001784850937	162358112844387525285	154390525937665500150	97805753994377172258	37147845781383864021	6408246094697758113				671124885227835500000
12	778700428129000000	17912534645834920000	188766377065523910000	1205038268809939336000	5189983425874442092300	15887146577044983727668	35442281775305802288321	58058268552952013087118	69308120872989492329115	58800696325170019520760	33652300542675663065607	11664923114551111628454	1851983121367652094657			291268200188880607000000
13	132379072781930000000	3309110334927667400000	38162633346077102580000	268848307522902292610000	1290805155525224590595000	4460219299469083102993260	11410930591420235872694022	21884839175830509200550879	31464133587858946132482552	33491571351222341218099185	25654327146487499338690830	13391166813440738756077953	4269246066465327698137596	627822278143634060088723		148255513896140228963000000
14	26078677338040210000000	703919711584048967800000	8818519130803762496460000	67961031487020052958110000	359946000494969845342945000	1385949628116822635393507220	4000819511201136126397073514	8795809040066763010477273809	14799176112909602503915558191	18963244056219367050031182381	18216089988888417468435073215	12720210430817412775052852931	6103770032775879887671741941	1801494692915169695205553659	246734155310448185614868139	87470753198722735088170000000

Technical details

Node id: n-e1501c7256f70865 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-b84972725564956e

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,             3;
  2:           10,            33,             27;
  3:          100,           520,            897,             513;
  4:         1700,         12140,          32409,           38322,           16929;
  5:        44200,        402340,        1461774,         2649231,         2394576,           863379;
  6:      1635400,      18113180,       83456458,       204731049,       281993175,        206749071,         63026667;
  7:     81770000,    1067563600,     5966027720,     18498741792,     34368032601,      38254777875,      23619491379,       6239640033;
  8:   5315050000,   79939964000,   525507506200,   1972035792360,   4620259810233,    6920036767404,    6470133285510,    3452490990036,    804913564257;
  9: 435834100000, 7421430198000, 56121829640400, 247364658484120, 700303138593786, 1320545363995107, 1658516922498672, 1337735986721082, 628758943644942, 131200910973891;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (n^2+1+(3*n^2+2)*x+(2*n^2+1)*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[((n^2 + 1 + (3*n^2 + 2)*x + (2*n^2 + 1)*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] = ((n^2 + 1 + (3*n^2 + 2)*x + (2*n^2 + 1)*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_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    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 central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ 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.