Real-rooted atlas

A111577 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 5\,x + 3\,n x\right)\,P_{n-1}(x) + \left(-3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31517
411221135
5122105851214
613532582034111523
7151780407060811365112349
8170159614210467814393254611112052
91922926397462315115117423129852184511120849
10111749509534086744735213855430405221274087381112219767
11114578752042402678907174005375120841056516790155765613495251143942992
12117611935401115717218768299770329617002722226120580744461109385452139810111819256321
131210173917352951720006222608788416273126325996098152996968402159168300627670962655592405124526654381
14124724531127441637788322656089434660124608036915038160105907254453135522722335599353968855375266260223696211350974470746
1512873367021084707729521817141624502300348193018214245192080729297949318300387935831821730787240604729235110376492334418947848515308470041299
Technical details
Node id
n-c9da60b665fe4f56 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c48676221a04fdaf

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   5,    1;
   4: 1,  12,   21,     1;
   5: 1,  22,  105,    85,      1;
   6: 1,  35,  325,   820,    341,       1;
   7: 1,  51,  780,  4070,   6081,    1365,       1;
   8: 1,  70, 1596, 14210,  46781,   43932,    5461,       1;
   9: 1,  92, 2926, 39746, 231511,  511742,  312985,   21845,     1;
  10: 1, 117, 4950, 95340, 867447, 3521385, 5430405, 2212740, 87381, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-1,x) - 3*x^2*d/dx 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[(1 + (3*n - 5)*x)*P[n - 1, x] - 3*x^2*D[P[n - 1, x], 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 + (3*n - 5)*x)*v[i - 1] - 3*x^2*deriv(v[i - 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. [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.