Real-rooted atlas

basis-map coefficients of A026729 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 6\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2156
31102536
411575125216
51201505006251296
61252501250312531257776
713037525009375187501562546656
81355254375218756562510937578125279936
91407007000437501750004375006250003906251679616
101459001050078750393750131250028125003515625195312510077696
11150112515000131250787500328125093750001757812519531250976562560466176
1215513752062520625014437507218750257812506445312510742187510742187548828125362797056
13160165027500309375247500014437500618750001933593754296875006445312505859375002441406252176782336
141651950357504468754021875268125001340625005027343751396484375279296875038085937503173828125122070312513060694016
15170227545500625625625625046921875268125000117304687539101562509775390625177734375002221679687517089843750610351562578364164096
Technical details
Node id
n-8b0b17ea58a3d398 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a8e18dd2f0709b33

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  5;
   3: 1, 10,  25;
   4: 1, 15,  75,   125;
   5: 1, 20, 150,   500,   625;
   6: 1, 25, 250,  1250,  3125,   3125;
   7: 1, 30, 375,  2500,  9375,  18750,   15625;
   8: 1, 35, 525,  4375, 21875,  65625,  109375,   78125;
   9: 1, 40, 700,  7000, 43750, 175000,  437500,  625000,  390625;
  10: 1, 45, 900, 10500, 78750, 393750, 1312500, 2812500, 3515625, 1953125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1+6*x+5*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 6*x + 5*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 6*x + 5*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, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.