A204420 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 - x - 2\,x^{2} - 4\,n - 2\,n x + 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	9	54	72													135
3	225	1710	3960	2880												8775
4	11025	99540	313740	418320	201600											1044225
5	893025	9253440	36163260	67767840	61508160	21772800										197358525
6	108056025	1257192090	5800784220	13769050680	17878734720	12106765440	3353011200									54273594375
7	18261468225	234941116410	1241828487900	3533532682680	5885468709120	5764820181120	3084866104320	697426329600								20461145079375
8	4108830350625	57792347613000	342845511208200	1130338545336000	2278284283875600	2886161092214400	2250596322374400	989834772326400	188305108992000							10128266814290625
9	1187451971330625	18098990779369500	118731750927589800	443235313412892000	1042739263454288400	1608717212167665600	1631717108519097600	1051264998809625600	390963999089664000	64023737057280000						6370679826188803125
10	428670161650355625	7030090595368590750	50427540230636368800	209637820029786548400	561701235113586968400	1016611925716419832800	1261493670861478454400	1061564415931257638400	580566773173792204800	186535520697157632000	26761922089943040000					4962759584601077634375
11	189043541287806830625	3316319714029327755750	25781710901776408378800	117865758909376597719600	353367705980104273458000	731423281738188978338400	1068691119410987594121600	1103942717539869759552000	791058412598996212070400	374867818307037786931200	105815910073032327168000	13488008733331292160000				4689807807448018364484375
12	100004033341249813400625	1867381166411622867505500	15621684256029258030323700	77768449582322512404190800	257415240291282366095602800	598236804215604325068897600	1002728724647849446336689600	1223062986986361684087724800	1078627645353711112397337600	671358006618622724104704000	280147571776242697656729600	70413070843605585199104000	8065829222532112711680000			5285413398993916696773890625
13	62502520838281133375390625	1237116052346139161571375000	11070719476506422276206162500	59540459968172050873845840000	215322439889677237492685310000	554088670838650359434982960000	1045471215855828931508659320000	1466324474119970664990510720000	1530286369236522624109743360000	1174638105884236981243576320000	645042836993187592908748800000	240111469520623379109150720000	54330292854606480084172800000	5646080455772478898176000000		7003172753666939623225405078125
14	45564337691106946230659765625	952484644039343166819598781250	9072618500773554560227106212500	52372278092767627130760608985000	205197831253794067339982721390000	578341817352014674397177678940000	1210960339738206082212148841880000	1915782226476680049300096364080000	2303301587210601231618316592640000	2095843138271192084855459258880000	1421693093934265710037774694400000	697525959245016393626657402880000	234097073802713061059774054400000	48123529864489385984950272000000	4573325169175707907522560000000	10777882867893420080143898415234375

Technical details

Node id: n-3df856e22293eef3 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-e426df4ea063f288

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:                1,                 4;
  2:                9,                54,                 72;
  3:              225,              1710,               3960,               2880;
  4:            11025,             99540,             313740,             418320,              201600;
  5:           893025,           9253440,           36163260,           67767840,            61508160,            21772800;
  6:        108056025,        1257192090,         5800784220,        13769050680,         17878734720,         12106765440,          3353011200;
  7:      18261468225,      234941116410,      1241828487900,      3533532682680,       5885468709120,       5764820181120,       3084866104320,        697426329600;
  8:    4108830350625,    57792347613000,    342845511208200,   1130338545336000,    2278284283875600,    2886161092214400,    2250596322374400,     989834772326400,    188305108992000;
  9: 1187451971330625, 18098990779369500, 118731750927589800, 443235313412892000, 1042739263454288400, 1608717212167665600, 1631717108519097600, 1051264998809625600, 390963999089664000, 64023737057280000;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (4*n^2-4*n+1+(8*n^2-2*n-1)*x+(4*n^2+2*n-2)*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[((4*n^2 - 4*n + 1 + (8*n^2 - 2*n - 1)*x + (4*n^2 + 2*n - 2)*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] = ((4*n^2 - 4*n + 1 + (8*n^2 - 2*n - 1)*x + (4*n^2 + 2*n - 2)*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 T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of e…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    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
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    ↑ Parent-row provers of polytool (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.