Real-rooted atlas

A038226 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 46\,x + 35\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 770\,x + 1225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012177012252116
1133112705404254287597336
214641186340889350188650015006254477456
3161051256217516304750518787508253437552521875205962976
417715613382071026902837511413325002723634375346644375018382656259474296896
5194871714340324454143036975219706506256990661562513345808437514154645312564339296875435817657216
62143588815456407880607645423003866834510001537945543750391477047500062280439375005661858125000225187539062520047612231936
72357947691675230475158593842411006380276941500304513217662509689056925625020552544993750028026197718750022293566367187578815638671875922190162669056

All 15 rows

Technical details
Node id
n-ef9b4cea55358f22 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-469248a0f2a08b50

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:          121,           770,            1225;
  1:         1331,         12705,           40425,            42875;
  2:        14641,        186340,          889350,          1886500,          1500625;
  3:       161051,       2562175,        16304750,         51878750,         82534375,          52521875;
  4:      1771561,      33820710,       269028375,       1141332500,       2723634375,        3466443750,         1838265625;
  5:     19487171,     434032445,      4143036975,      21970650625,      69906615625,      133458084375,       141546453125,        64339296875;
  6:    214358881,    5456407880,     60764542300,     386683451000,    1537945543750,     3914770475000,      6228043937500,      5661858125000,      2251875390625;
  7:   2357947691,   67523047515,    859384241100,    6380276941500,   30451321766250,    96890569256250,    205525449937500,    280261977187500,    222935663671875,     78815638671875;
  8:  25937424601,  825281691850,  11816533315125,  100261494795000,  558274232381250,  2131592523637500,   5651949873281250,  10276272496875000,  12261461501953125,   8669720253906250,   2758547353515625;
  9: 285311670611, 9985908471385, 158866725681125, 1516455108774375, 9650168874018750, 42987115893356250, 136777186933406250, 310857243030468750, 494545613912109375, 524518075361328125, 333784229775390625, 96549157373046875;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+46*x+35*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 770*x + 1225*x^2.
%t Mathematica
P[0, x_] := 121 + 770*x + 1225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 46*x + 35*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 770*x + 1225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 46*x + 35*x^2)*v[i - 1])/(1 + x)); 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. [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.