A272866 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 40\,x + 106\,x^{2} + 112\,x^{3} + 41\,x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	5	30	41																											76
2	25	300	1310	2460	1681																									5776
3	125	2250	16575	63900	135915	151290	68921																							438976
4	625	15000	155500	909000	3276150	7453800	10455820	8270520	2825761																					33362176
5	3125	93750	1253125	9825000	50026250	172822500	410215250	660633000	690933025	423864150	115856201																			2535525376
6	15625	562500	9206250	90562500	596259375	2767725000	9286827500	22695345000	40092480375	49933264500	41623459530	20854116180	4750104241																	192699928576
7	78125	3281250	63546875	752062500	6075628125	35439468750	153912521875	505558275000	1262082679375	2382949878750	3349906928025	3400238211300	2355935847335	997521890610	194754273881															14645194571776
8	390625	18750000	419375000	5801250000	55545437500	390300750000	2081847425000	8598185250000	27787575043750	70505119050000	139983420857000	215199343926000	251133009624700	215075451536400	127492797828440	46741025731440	7984925229121													1113034787454976
9	1953125	105468750	2675390625	42356250000	468959062500	3855718125000	24395622562500	121448679750000	482239177143750	1538678441812500	3954361252578750	8166209226390000	13450965621040500	17432540647389000	17386180929846900	12876582576502800	6669360109054845	2155929811862670	327381934393961											84590643846578176
10	9765625	585937500	16621093750	296367187500	3725173828125	35083968750000	256876978125000	1497196518750000	7054876803281250	27139963393125000	85703965780162500	222547699823625000	474369916252631250	825506250150150000	1161396714533565000	1300702507320870000	1132477695964081125	738800337967573500	339758568499098550	98214580318188300	13422659310152401									6428888932339941376
11	48828125	3222656250	101083984375	2004492187500	28198349609375	299326113281250	2489636080078125	16630734656250000	90722235682031250	408731178332812500	1531968679629093750	4796595971641125000	12562143172958568750	27483084431098312500	50021337643530206250	75191170228767450000	92300728995322670625	90997135368915971250	70294388516050023875	40974643813234411500	16943652014559451555	4429477572350292330	550329031716248441							488595558857835544576
12	244140625	17578125000	604101562500	13187109375000	205270957031250	2424765234375000	22584096132812500	170115126328125000	1054608297380859375	5447183083031250000	23641390411093125000	86700018558723750000	269519310878819437500	710940152181534750000	1589647091241901725000	3003402442124774250000	4768113595224375309375	6306845526083725425000	6865715892455958752500	6044587424673227595000	4196027503627880581650	2210417344641145879800	830325704926027525860	198118451417849438760	22563490300366186081					37133262473195501387776
13	1220703125	95214843750	3557861328125	84779296875000	1446736230468750	18822626367187500	194079546933593750	1627513890234375000	11302443218193359375	65848884516035156250	324861384739018359375	1366076311530693750000	4917894118010627812500	15194980848179930625000	40326731767687148062500	91854971187323847750000	179118171980783074771875	297717135614265630656250	419026602648990944196875	494775080466477642975000	483812111849030539610750	384760996243644478243500	242501276512607617870950	116527474535157054041400	40099724896004442653465	8799761217142812571590	925103102315013629321			2822127947962858105470976
14	6103515625	512695312500	20695800781250	534536132812500	9922932373046875	140991169921875000	1594392711132812500	14731683540234375000	113294894222275390625	735045788625585937500	4063173631122074218750	19276027364981460937500	78891076710273704296875	279520856553976931250000	859116743122269399375000	2292071023742610836250000	5304635997998803876921875	10628184656175098154187500	18370492132885029706318750	27251076041987007397687500	34442403681848110699375625	36723946635814721941515000	32591632855682624560700500	23632876514043437414319000	13638875197551646167218275	6024617008907286558159300	1912707072762041594086370	388543302972305724314820	37929227194915558802161	214481724045177216015794176

Technical details

Node id: n-65ae6116fb370fbc · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-158304e21352a080

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:       1;
  1:       5,        30,         41;
  2:      25,       300,       1310,        2460,         1681;
  3:     125,      2250,      16575,       63900,       135915,        151290,          68921;
  4:     625,     15000,     155500,      909000,      3276150,       7453800,       10455820,         8270520,         2825761;
  5:    3125,     93750,    1253125,     9825000,     50026250,     172822500,      410215250,       660633000,       690933025,        423864150,        115856201;
  6:   15625,    562500,    9206250,    90562500,    596259375,    2767725000,     9286827500,     22695345000,     40092480375,      49933264500,      41623459530,      20854116180,        4750104241;
  7:   78125,   3281250,   63546875,   752062500,   6075628125,   35439468750,   153912521875,    505558275000,   1262082679375,    2382949878750,    3349906928025,    3400238211300,     2355935847335,      997521890610,      194754273881;
  8:  390625,  18750000,  419375000,  5801250000,  55545437500,  390300750000,  2081847425000,   8598185250000,  27787575043750,   70505119050000,  139983420857000,  215199343926000,   251133009624700,   215075451536400,   127492797828440,    46741025731440,    7984925229121;
  9: 1953125, 105468750, 2675390625, 42356250000, 468959062500, 3855718125000, 24395622562500, 121448679750000, 482239177143750, 1538678441812500, 3954361252578750, 8166209226390000, 13450965621040500, 17432540647389000, 17386180929846900, 12876582576502800, 6669360109054845, 2155929811862670, 327381934393961;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+40*x+106*x^2+112*x^3+41*x^4)*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[((5 + 40*x + 106*x^2 + 112*x^3 + 41*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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] = ((5 + 40*x + 106*x^2 + 112*x^3 + 41*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    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
    
    ↑ 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.