Real-rooted atlas

A013612 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 27\,x + 21\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 252\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
036252441729
121622687938926119683
212961814495256222264194481531441
3777613608095256033339605834430408410114348907
44665697977685730404000752010501974014702763685766121387420489
5279936685843272013536420078960147027636030875803563602177082180108854110460353203
616796164702924857610828840327580161764331632049401285696864522499688645224996837822859361282429536481
7100776963174474244444263936362948221441905478162566669173568961556140499424233421074913620424344054947942800465817625597484987

All 15 rows

Technical details
Node id
n-72a51ec1ec4c0613 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8d10fbbacee008e8

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:        36,         252,          441;
  1:       216,        2268,         7938,          9261;
  2:      1296,       18144,        95256,        222264,         194481;
  3:      7776,      136080,       952560,       3333960,        5834430,        4084101;
  4:     46656,      979776,      8573040,      40007520,      105019740,      147027636,        85766121;
  5:    279936,     6858432,     72013536,     420078960,     1470276360,     3087580356,      3602177082,      1801088541;
  6:   1679616,    47029248,    576108288,    4032758016,    17643316320,    49401285696,     86452249968,     86452249968,      37822859361;
  7:  10077696,   317447424,   4444263936,   36294822144,   190547816256,   666917356896,   1556140499424,   2334210749136,    2042434405494,     794280046581;
  8:  60466176,  2116316160,  33331979520,  311098475520,  1905478162560,  8003008282752,  23342107491360,  46684214982720,   61273032164820,   47656802794860,   16679880978201;
  9: 362797056, 13967686656, 244434516480, 2566562423040, 17965936961280, 88033091110272, 308115818885952, 770289547214880, 1348006707626040, 1572674492230380, 1100872144561266, 350277500542221;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (6+27*x+21*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 252*x + 441*x^2.
%t Mathematica
P[0, x_] := 36 + 252*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 27*x + 21*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] = 36 + 252*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 27*x + 21*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

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.