Real-rooted atlas

A038314 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(46 + 48\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2116 + 4416\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02116441623048836
197336304704317952110592830584
24477456186885122925158420348928530841678074896
320596297610745894402242621440234012672012209356802548039687339040224
49474296896593173370881547408793602152916582401684891238407032589516812230590464689869781056
543581765721631833637570569965312630784173309784883201808449929216011322469122048393825012940858706834227264847759419264
620047612231936167353980370944611205841354752127556001674035216637739348787201388889545637888724638023811072216041149956096281792804290566095689385410816
79221901626690568660568484196352361484597601239048801364115508428813776048180795801614375006797352140810000004728592793644720518040911872116662220976291841352605460594688572994802228616704

All 15 rows

Technical details
Node id
n-f7d94fed0ec64a12 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-723ac5bf98d71a32

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:                2116,                 4416,                  2304;
  1:               97336,               304704,                317952,                110592;
  2:             4477456,             18688512,              29251584,              20348928,               5308416;
  3:           205962976,           1074589440,            2242621440,            2340126720,            1220935680,              254803968;
  4:          9474296896,          59317337088,          154740879360,          215291658240,          168489123840,            70325895168,            12230590464;
  5:        435817657216,        3183363757056,         9965312630784,        17330978488320,        18084499292160,         11322469122048,          3938250129408,          587068342272;
  6:      20047612231936,      167353980370944,       611205841354752,      1275560016740352,      1663773934878720,       1388889545637888,        724638023811072,       216041149956096,        28179280429056;
  7:     922190162669056,     8660568484196352,     36148459760123904,     88013641155084288,    137760481807958016,     143750067973521408,     100000047285927936,     44720518040911872,     11666222097629184,      1352605460594688;
  8:   42420747482776576,   442651278081146880,   2078536436207124480,   5783753561619824640,  10561636938610114560,   13225006253563969536,   11500005437881712640,   6857146099606487040,   2683231082454712320,    622198511873556480,    64925062108545024;
  9: 1951354384207722496, 22398154670906032128, 116859937413422776320, 365822412772453908480, 763455470133816852480, 1115308860717228097536, 1163800550313629319168, 867428981600220610560, 452571642574028144640, 157416223504009789440, 32852081426923782144, 3116402981210161152;
%F Formula
The row polynomials satisfy P(n,x) = (46+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 2116 + 4416*x + 2304*x^2.
%t Mathematica
P[0, x_] := 2116 + 4416*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(46 + 48*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] = 2116 + 4416*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (46 + 48*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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.