Real-rooted atlas

A167884 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 8\,n\right)\,P_{n}(x) = \left(7 - 210\,x - 105\,x^{2} + 56\,n - 1568\,n x - 664\,n x^{2} + 896\,n^{2} x + 1344\,n^{2} x^{2} - 512\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(56\,x - 56\,x^{3} + 448\,n x - 64\,n x^{2} - 384\,n x^{3} - 512\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-14 + 98\,x - 48\,n - 224\,n x + 128\,n^{2} x\right)\,P_{n-1}(x) + \left(-56\,x + 56\,x^{2} + 64\,n x - 64\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2136340
313585374900
413272182586544528080
51295144330865774487378561113840
6126570888234052887240014705675797124753464320
71239147016633892711100398201387549775332677854837014583007368000

All 15 rows

Technical details
Node id
n-87466fb132feaebb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2da1e39d3dcaff09

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,         2;
  2: 1,        36,           3;
  3: 1,       358,         537,             4;
  4: 1,      3272,       18258,          6544,             5;
  5: 1,     29514,      433086,        577448,         73785,             6;
  6: 1,    265708,     8823405,      28872400,      14705675,        797124,             7;
  7: 1,   2391470,   166338927,    1110039820,    1387549775,     332677854,       8370145,            8;
  8: 1,  21523344,  3003534804,   36844190176,   91578285150,   55266285264,    7008247876,     86093376,         9;
  9: 1, 193710226, 52900337580, 1117335694928, 4910348156470, 5892417787764, 1955337466124, 141067566880, 871696017, 10;
%F Formula
The row polynomials satisfy (8*n-7)*P(n,x) = (56*n+7+(896*n^2-1568*n-210)*x-(512*n^3-1344*n^2+664*n+105)*x^2)*P(n-2,x) + ((448*n+56)*x-(512*n^2+64*n)*x^2+(512*n^2-384*n-56)*x^3)*d/dx P(n-2,x) - (48*n+14-(128*n^2-224*n+98)*x)*P(n-1,x) + ((64*n-56)*x-(64*n-56)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((56*n + 7 + (896*n^2 - 1568*n - 210)*x - (512*n^3 - 1344*n^2 + 664*n + 105)*x^2)*P[n - 2, x] + ((448*n + 56)*x - (512*n^2 + 64*n)*x^2 + (512*n^2 - 384*n - 56)*x^3)*D[P[n - 2, x], x] - (48*n + 14 - (128*n^2 - 224*n + 98)*x)*P[n - 1, x] + ((64*n - 56)*x - (64*n - 56)*x^2)*D[P[n - 1, x], x])/(8*n - 7)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((56*n + 7 + (896*n^2 - 1568*n - 210)*x - (512*n^3 - 1344*n^2 + 664*n + 105)*x^2)*v[i - 2] + ((448*n + 56)*x - (512*n^2 + 64*n)*x^2 + (512*n^2 - 384*n - 56)*x^3)*deriv(v[i - 2], x) - (48*n + 14 - (128*n^2 - 224*n + 98)*x)*v[i - 1] + ((64*n - 56)*x - (64*n - 56)*x^2)*deriv(v[i - 1], x))/(8*n - 7)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.