Real-rooted atlas

A027465 → basis A013611 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181624
264256256576
351230726144409613824
44096327689830413107265536331776
53276832768013107202621440262144010485767962624
626214431457281572864041943040629145605033164816777216191102976
7209715229360128176160768587202560117440512014092861449395240962684354564586471424
81677721626843545618790481927516192768187904819203006477107230064771072171798691844294967296110075314176
913421772824159191041932735283290194313216270582939648541165879296721554505728618475290624309237645312687194767362641807540224
101073741824214748364801932735283201030792151040360777252864086586540687361443109011456016492674416640123695058124805497558138880109951162777663403380965376
1185899345921889785610241889785610240113387136614404535485464576012699359300812825398718601625636283883716608036283883716608024189255811072096757023244288175921860444161521681143169024
1268719476736164926744166418141941858304120946279055360544258255749120174162641839718440637949762600966966505673588736870813209198592077405618595430404644337115725824168884986026393628147497671065636520347436056576
135497558138881429365116108817152381393305612578413021757446289206510878720226411434391633926037638250443571212075276500887142418112914751330713620125460834811904016100368667849523287820192733724672292733975779082244503599627370496876488338465357824
144398046511104123145302310912160088893004185612807111440334848704391129218416642817564516873666568452693550620999681932044240141942784338107742024839987245081032269978664964508103226997866496327862052872572108816393102643628605445044031582654955527205759403792793621035720123168587776
Technical details
Node id
n-775b9e7cf85328d0 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-11aec07ce60b6d61

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:         8,         16;
  2:        64,        256,         256;
  3:       512,       3072,        6144,        4096;
  4:      4096,      32768,       98304,      131072,        65536;
  5:     32768,     327680,     1310720,     2621440,      2621440,      1048576;
  6:    262144,    3145728,    15728640,    41943040,     62914560,     50331648,     16777216;
  7:   2097152,   29360128,   176160768,   587202560,   1174405120,   1409286144,    939524096,    268435456;
  8:  16777216,  268435456,  1879048192,  7516192768,  18790481920,  30064771072,  30064771072,  17179869184,   4294967296;
  9: 134217728, 2415919104, 19327352832, 90194313216, 270582939648, 541165879296, 721554505728, 618475290624, 309237645312, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (8+16*x)*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[(8 + 16*x)*P[n - 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] = (8 + 16*x)*v[i - 1]); 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

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.