Real-rooted atlas

A188440 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(18 + 9\,x + 6\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(-18 - 69\,x - 36\,x^{2} - 4\,x^{3} + 36\,n + 66\,n x + 11\,n x^{2} - 2\,n x^{3} + 18\,n^{2} x + 25\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-54 - 162\,x - 156\,x^{2} - 60\,x^{3} - 8\,x^{4} - 36\,n x - 84\,n x^{2} - 58\,n x^{3} - 12\,n x^{4} - 6\,n^{2} x^{2} - 10\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2347
3347
49241245
59241245
62710810832275
72710810832275
881432648384801625
981432648384801625
1024316203240288012001929375
1124316203240288012001929375
127295832145801728010800345644853125
137295832145801728010800345644853125
142187204126123690720756003628894081024296875
Technical details
Node id
n-02cb9b0377f6834a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ee9e93d4fe03de68

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  1;
  2:  3,   4;
  3:  3,   4;
  4:  9,  24,  12;
  5:  9,  24,  12;
  6: 27, 108, 108,  32;
  7: 27, 108, 108,  32;
  8: 81, 432, 648, 384, 80;
  9: 81, 432, 648, 384, 80;
%F Formula
The row polynomials satisfy (18+(6*n+9)*x+(n^2+n)*x^2)*P(n,x) = (36*n-18+(18*n^2+66*n-69)*x+(2*n^3+25*n^2+11*n-36)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (54+(36*n+162)*x+(6*n^2+84*n+156)*x^2+(10*n^2+58*n+60)*x^3+(4*n^2+12*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((36*n - 18 + (18*n^2 + 66*n - 69)*x + (2*n^3 + 25*n^2 + 11*n - 36)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (54 + (36*n + 162)*x + (6*n^2 + 84*n + 156)*x^2 + (10*n^2 + 58*n + 60)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(18 + (6*n + 9)*x + (n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((36*n - 18 + (18*n^2 + 66*n - 69)*x + (2*n^3 + 25*n^2 + 11*n - 36)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (54 + (36*n + 162)*x + (6*n^2 + 84*n + 156)*x^2 + (10*n^2 + 58*n + 60)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(18 + (6*n + 9)*x + (n^2 + n)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.