Real-rooted atlas

A102537 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-4\,n - 6\,n x + 22\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-5 - 8\,x - 3\,x^{2} + 17\,n + 26\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
321321265
412629022055691
58182520288614562737953
655942159533110252009520142896447
7396931841923540083606722054286201677521213761
8289510156933036269104632390353115016063744037884326315702715

All 15 rows

Technical details
Node id
n-c37007757d9d9c3d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ba125e795100165f

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:        4,         3;
   3:       21,        32,        12;
   4:      126,       290,       220,        55;
   5:      818,      2520,      2886,      1456,       273;
   6:     5594,     21595,     33110,     25200,      9520,      1428;
   7:    39693,    184192,    354008,    360672,    205428,     62016,      7752;
   8:   289510,   1569330,   3626910,   4632390,   3531150,   1606374,    403788,    43263;
   9:  2157150,  13376480,  36124200,  55489280,  53023740,  32274704,  12219900,  2631200,   246675;
  10: 16348960, 114139278, 352716408, 633200304, 727714020, 555216090, 281209500, 91171080, 17168580, 1430715;
%F Formula
The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (22*n^2-4*n+(18*n^2-6*n)*x)*P(n-1,x) + (17*n-5+(26*n-8)*x+(9*n-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[Together[((22*n^2 - 4*n + (18*n^2 - 6*n)*x)*P[n - 1, x] + (17*n - 5 + (26*n - 8)*x + (9*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
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] = ((22*n^2 - 4*n + (18*n^2 - 6*n)*x)*v[i - 1] + (17*n - 5 + (26*n - 8)*x + (9*n - 3)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); 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.