A272866 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 30\,x + 64\,x^{2} + 58\,x^{3} + 19\,x^{4}\right)\,P_{n-1}(x)\)

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

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	20	19																											44
2	25	200	590	760	361																									1936
3	125	1500	7425	19400	28215	21660	6859																							85184
4	625	10000	69500	274000	670150	1041200	1003580	548720	130321																					3748096
5	3125	62500	559375	2950000	10151250	23815000	38574750	42598000	30694025	13032100	2476099																			164916224
6	15625	375000	4106250	27125000	120384375	378150000	862047500	1436970000	1738350375	1488403000	856208970	297131880	47045881																	7256313856
7	78125	2187500	28328125	224875000	1222440625	4813812500	14160540625	31610650000	53810054375	69511452500	67077761975	46889495800	22445837435	6586423340	893871739															319277809664
8	390625	12500000	186875000	1732500000	11147937500	52790500000	190305325000	532726500000	1170313543750	2024360700000	2748008893000	2896720316000	2324496580700	1372749285600	562668736760	143019478240	16983563041													14048223625216
9	1953125	70312500	1191796875	12637500000	93940312500	519828750000	2219147437500	7472758500000	20121898893750	43649877875000	76463215796250	107906632740000	121769058190500	108391364046000	74433764429100	38050708552800	13636013378445	3057041347380	322687697779											618121839509504
10	9765625	390625000	7402343750	88359375000	745095703125	4718062500000	23277178125000	91623487500000	292228465781250	762669778750000	1637629715462500	2898145159250000	4219779045881250	5027564006100000	4853608208685000	3738364748220000	2243435762101125	1010968936809000	321838519626950	64537539555800	6131066257801									27197360938418176
11	48828125	2148437500	45009765625	597265625000	5633310546875	40171054687500	224903958984375	1013304187500000	3735878463281250	11398764471875000	28993885002156250	61734045901750000	110176763008193750	164598158973875000	204995123037168750	211287704030700000	178203029739920625	120952490142247500	64453850814236125	25967867889689000	7436337995317055	1348834576716220	116490258898219							1196683881290399744
12	244140625	11718750000	268945312500	3927343750000	40967050781250	324869531250000	2034973789062500	10327850156250000	43218651287109375	150964171187500000	443926405250625000	1105124454517500000	2336448548116937500	4199472927166500000	6410297291819025000	8283706001400500000	9011676567019809375	8183289422092950000	6127176621974622500	3717013668435870000	1781162213240851650	648860422694013200	168849564739839540	27957662135572560	2213314919066161					52654090776777588736
13	1220703125	63476562500	1583740234375	25238281250000	288492089843750	2518308203125000	17450633535156250	98511247656250000	461314761552734375	1815423034648437500	6060069824458203125	17272469630162500000	42219333530696562500	88723700234003750000	160433467416646937500	249414461459546500000	332528151407670521875	378540392477470437500	365523526325108490625	296611099801437850000	199662440252439485750	109490806078568629000	47663538329155643050	15845127636660904400	3778361547363733265	575461878957201860	42052983462257059			2316779994178213904384
14	6103515625	341796875000	9211425781250	159072265625000	1977317138671875	18840910156250000	143100681445312500	889416764843750000	4608360798056640625	20175124109765625000	75373790284761718750	242056782298296875000	671687386921438671875	1616182094757037500000	3378808678846544375000	6141491960076742500000	9699165867145574421875	13282139758272146125000	15716460357920691118750	15985793482581297375000	13875481197468015750625	10176313731668952010000	6221720178013501024500	3112821720259221334000	1241401588723993972275	379501965438617858200	83508371896366254530	11774835369431976520	799006685782884121	101938319743841411792896

Technical details

Node id: n-8253cbfd755ffdf6 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-a634a8e2c7c6279b

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,       20,         19;
  2:      25,      200,        590,         760,         361;
  3:     125,     1500,       7425,       19400,       28215,        21660,          6859;
  4:     625,    10000,      69500,      274000,      670150,      1041200,       1003580,        548720,         130321;
  5:    3125,    62500,     559375,     2950000,    10151250,     23815000,      38574750,      42598000,       30694025,       13032100,        2476099;
  6:   15625,   375000,    4106250,    27125000,   120384375,    378150000,     862047500,    1436970000,     1738350375,     1488403000,      856208970,       297131880,        47045881;
  7:   78125,  2187500,   28328125,   224875000,  1222440625,   4813812500,   14160540625,   31610650000,    53810054375,    69511452500,    67077761975,     46889495800,     22445837435,      6586423340,      893871739;
  8:  390625, 12500000,  186875000,  1732500000, 11147937500,  52790500000,  190305325000,  532726500000,  1170313543750,  2024360700000,  2748008893000,   2896720316000,   2324496580700,   1372749285600,   562668736760,   143019478240,    16983563041;
  9: 1953125, 70312500, 1191796875, 12637500000, 93940312500, 519828750000, 2219147437500, 7472758500000, 20121898893750, 43649877875000, 76463215796250, 107906632740000, 121769058190500, 108391364046000, 74433764429100, 38050708552800, 13636013378445, 3057041347380, 322687697779;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+30*x+64*x^2+58*x^3+19*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 + 30*x + 64*x^2 + 58*x^3 + 19*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 + 30*x + 64*x^2 + 58*x^3 + 19*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_A013609 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.