Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 12\,x + 46\,x^{2} + 60\,x^{3} + 25\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11102536
21201505006251296
313037525009375187501562546656
41407007000437501750004375006250003906251679616
5150112515000131250787500328125093750001757812519531250976562560466176
6160165027500309375247500014437500618750001933593754296875006445312505859375002441406252176782336
7170227545500625625625625046921875268125000117304687539101562509775390625177734375002221679687517089843750610351562578364164096
8180300070000113750013650000125125000893750000502734375022343750000782031250002132812500004443359375006835937500007324218750004882812500001525878906252821109907456
91903825102000191250026775000290062500248625000017092968750949609375004273242187501553906250000453222656250010458984375000186767578125002490234375000023345947265625137329101562503814697265625101559956668416
1011004750142500302812548450000605625000605625000049207031250328046875000180425781250082011718750003075439453125094628906250000236572265625000473144531250000739288330078125869750976562500724792480468750381469726562500953674316406253656158440062976
1111105775192500457187582293750116582812513323750000124910156250971523437500631490234375034444921875000157872558593750607202148437500195172119140625052045898437500001138504028320312520091247558593750279045104980468752937316894531250022029876708984375104904174804687502384185791015625131621703842267136
12112069002530006641250132825000210306250027039375000287293359375255371875000019152890625000121882031250000660194335937500304705078125000011970556640625000399018554687500001122239685058593752640563964843750005134429931640625008106994628906250001013374328613281250965118408203125000658035278320312500286102294921875000596046447753906254738381338321616896
13113081253250009343750205562500359734375051390625000610263671875610263671875051872412109375377253906250000235783691406250012696044921875000589459228515625002357836914062500008105064392089843752383842468261718750595960617065429687512546539306640625000219564437866210937503136634826660156250035643577575683593750309944152832031250001937150955200195312577486038208007812501490116119384765625170581728179578208256
141140945040950012796875307125000588656250092503125000121410351562513490039062500128155371093750104854394531250074271862792968754570576171875000024485229492187500011426440429687500004641991424560546875163834991455078125005006069183349609375013173866271972656250029641199111938476562556459426879882812500089821815490722656250011715888977050781250001220405101776123046875976324081420898437500563263893127441406250208616256713867187500372529029846191406256140942214464815497216
Technical details
Node id
n-78c39f1a29a465e8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-83d381845d6677d6

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1, 10,   25;
  2: 1, 20,  150,    500,     625;
  3: 1, 30,  375,   2500,    9375,    18750,     15625;
  4: 1, 40,  700,   7000,   43750,   175000,    437500,     625000,      390625;
  5: 1, 50, 1125,  15000,  131250,   787500,   3281250,    9375000,    17578125,    19531250,      9765625;
  6: 1, 60, 1650,  27500,  309375,  2475000,  14437500,   61875000,   193359375,   429687500,    644531250,     585937500,     244140625;
  7: 1, 70, 2275,  45500,  625625,  6256250,  46921875,  268125000,  1173046875,  3910156250,   9775390625,   17773437500,   22216796875,    17089843750,     6103515625;
  8: 1, 80, 3000,  70000, 1137500, 13650000, 125125000,  893750000,  5027343750, 22343750000,  78203125000,  213281250000,  444335937500,   683593750000,   732421875000,   488281250000,   152587890625;
  9: 1, 90, 3825, 102000, 1912500, 26775000, 290062500, 2486250000, 17092968750, 94960937500, 427324218750, 1553906250000, 4532226562500, 10458984375000, 18676757812500, 24902343750000, 23345947265625, 13732910156250, 3814697265625;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+12*x+46*x^2+60*x^3+25*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[((1 + 12*x + 46*x^2 + 60*x^3 + 25*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] = ((1 + 12*x + 46*x^2 + 60*x^3 + 25*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

Proof chains ? JSON

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.