A272866 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 35\,x + 84\,x^{2} + 83\,x^{3} + 29\,x^{4}\right)\,P_{n-1}(x)\)

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

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	25	29																											59
2	25	250	915	1450	841																									3481
3	125	1875	11550	37375	66990	63075	24389																							205379
4	625	12500	108250	530000	1604275	3074000	3641530	2438900	707281																					12117361
5	3125	78125	871875	5718750	24410625	70846875	141581625	192378750	170113275	88410125	20511149																			714924299
6	15625	468750	6403125	52656250	290306250	1130343750	3186988125	6555993750	9765902250	10273866250	7246093845	3076672350	594823321																	42180533641
7	78125	2734375	44187500	436953125	2953628125	14436406250	52612415625	145234640625	305152010625	485640706250	576288290725	494477829125	290027646860	104094081175	17249876309															2488651484819
8	390625	15625000	291562500	3368750000	26973406250	158694375000	709627450000	2460139375000	6682386121875	14268808375000	23871867418000	30963176895000	30524444393450	22111018622000	11099403169860	3449975261800	500246412961													146830437604321
9	1953125	87890625	1859765625	24585937500	227541093750	1565500781250	8297725406250	34643520000000	115494611034375	309747736796875	669868743999375	1165408012800000	1618985799464250	1771598332901250	1493481368809350	935954495593500	410633305535745	112555442916225	14507145975869											8662995818654939
10	9765625	488281250	11552734375	171972656250	1806287109375	14229023437500	87224838281250	426060257812500	1684195091953125	5440766039843750	14442380859915625	31556443031093750	56656322893303125	83129469022312500	98707953351041250	93393158352337500	68762988612987375	37971290225186250	14794787663321575	3626786493967250	420707233300201									511116753300641401
11	48828125	2685546875	70253906250	1162841796875	13665781250000	121289501953125	844232103515625	4723563925781250	21602002215234375	81664454974609375	257072712962312500	676623951808828125	1491021735181412500	2747192265345859375	4214809856218809375	5345419227184781250	5541174549053570625	4617332758668403125	3017382863722898000	1489171040583276875	521822040752007930	115694489157555275	12200509765705829							30155888444737842659
12	244140625	14648437500	419824218750	7648437500000	99437314453125	981814453125000	7649705722656250	48241017773437500	250581840222656250	1085355684101562500	3953433003418593750	12178206777365625000	31825816147034828125	70633599308720625000	132993486235001493750	211765918236424062500	283570839255232856250	316633185108039937500	291214295207878116250	216783076797372525000	127342489266306003525	56809983641503004000	18086203959575640990	3660152929711748700	353814783205469041					1779197418239532716881
13	1220703125	79345703125	2472412109375	49162597656250	700572412109375	7616809814453125	65677572060546875	460920351074218750	2680776111406250000	13090313941503906250	54167930486089062500	191202173811238281250	578233807268453984375	1501981396763076953125	3353756082157033109375	6432041127010055781250	10568813253001809162500	14813648535777318906250	17595455442142141850000	17546635132316973718750	14501503927294308513875	9754321376988968333125	5203612799558327407175	2117945389241536884250	617772811986514651415	114989804541777438325	10260628712958602189			104972647676132430295979
14	6103515625	427246093750	14381103515625	309924316406250	4803626953125000	57024074707031250	539124705615234375	4167428541503906250	26831538923642578125	145837662673828125000	675840008198824218750	2689828235512031250000	9242093365719809765625	27510615206052644531250	71087095740647254921875	159561568195105338281250	310904020822814400515625	524817766687223441250000	764814074942196147618750	957215369252680663125000	1021441605766517999188125	920161623904494792356250	690419712314050792266375	423555366867845951261250	206942269978162490171400	77439685610376323120250	20841459804718153860105	3591220049535510766150	297558232675799463481	6193386212891813387462761

Technical details

Node id: n-89fa4fd1d325b86a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8c53afe4f9846d4c

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,       25,         29;
  2:      25,      250,        915,        1450,          841;
  3:     125,     1875,      11550,       37375,        66990,         63075,         24389;
  4:     625,    12500,     108250,      530000,      1604275,       3074000,       3641530,        2438900,          707281;
  5:    3125,    78125,     871875,     5718750,     24410625,      70846875,     141581625,      192378750,       170113275,        88410125,        20511149;
  6:   15625,   468750,    6403125,    52656250,    290306250,    1130343750,    3186988125,     6555993750,      9765902250,     10273866250,      7246093845,       3076672350,        594823321;
  7:   78125,  2734375,   44187500,   436953125,   2953628125,   14436406250,   52612415625,   145234640625,    305152010625,    485640706250,    576288290725,     494477829125,     290027646860,     104094081175,      17249876309;
  8:  390625, 15625000,  291562500,  3368750000,  26973406250,  158694375000,  709627450000,  2460139375000,   6682386121875,  14268808375000,  23871867418000,   30963176895000,   30524444393450,   22111018622000,   11099403169860,   3449975261800,    500246412961;
  9: 1953125, 87890625, 1859765625, 24585937500, 227541093750, 1565500781250, 8297725406250, 34643520000000, 115494611034375, 309747736796875, 669868743999375, 1165408012800000, 1618985799464250, 1771598332901250, 1493481368809350, 935954495593500, 410633305535745, 112555442916225, 14507145975869;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+35*x+84*x^2+83*x^3+29*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 + 35*x + 84*x^2 + 83*x^3 + 29*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 + 35*x + 84*x^2 + 83*x^3 + 29*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_A013610 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.