Real-rooted atlas

A127773 → (θ+1)² → x ↦ x+1

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4\,n^{2} - 12\,n^{3} + 13\,n^{4} - 6\,n^{5} + n^{6}\right)\,P_{n}(x) = \left(n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3} - 2\,n^{4} - 6\,n^{4} x - 6\,n^{4} x^{2} - 2\,n^{4} x^{3} + n^{6} + 3\,n^{6} x + 3\,n^{6} x^{2} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(-12\,n^{2} - 24\,n^{2} x - 12\,n^{2} x^{2} - 12\,n^{3} - 24\,n^{3} x - 12\,n^{3} x^{2} + 9\,n^{4} + 18\,n^{4} x + 9\,n^{4} x^{2} + 6\,n^{5} + 12\,n^{5} x + 6\,n^{5} x^{2} - 3\,n^{6} - 6\,n^{6} x - 3\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(12 + 12\,x - 12\,n - 12\,n x - 21\,n^{2} - 21\,n^{2} x + 24\,n^{3} + 24\,n^{3} x + 6\,n^{4} + 6\,n^{4} x - 12\,n^{5} - 12\,n^{5} x + 3\,n^{6} + 3\,n^{6} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12 + 12\,x\)
\(P_{2}(x) = 54 + 108\,x + 54\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1121224
25410854216
31604804801601280
43751500225015003756000
5756378075607560378075624192
6137282322058027440205808232137287808
723041612848384806408064048384161282304294912

All 15 rows

Technical details
Node id
n-52a557efbe1a7c54 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:   12,    12;
  2:   54,   108,     54;
  3:  160,   480,    480,    160;
  4:  375,  1500,   2250,   1500,    375;
  5:  756,  3780,   7560,   7560,   3780,    756;
  6: 1372,  8232,  20580,  27440,  20580,   8232,   1372;
  7: 2304, 16128,  48384,  80640,  80640,  48384,  16128,   2304;
  8: 3645, 29160, 102060, 204120, 255150, 204120, 102060,  29160,  3645;
  9: 5500, 49500, 198000, 462000, 693000, 693000, 462000, 198000, 49500, 5500;
%F Formula
The row polynomials satisfy (n^6-6*n^5+13*n^4-12*n^3+4*n^2)*P(n,x) = (n^6-2*n^4+n^2+(3*n^6-6*n^4+3*n^2)*x+(3*n^6-6*n^4+3*n^2)*x^2+(n^6-2*n^4+n^2)*x^3)*P(n-3,x) - (3*n^6-6*n^5-9*n^4+12*n^3+12*n^2+(6*n^6-12*n^5-18*n^4+24*n^3+24*n^2)*x+(3*n^6-6*n^5-9*n^4+12*n^3+12*n^2)*x^2)*P(n-2,x) + (3*n^6-12*n^5+6*n^4+24*n^3-21*n^2-12*n+12+(3*n^6-12*n^5+6*n^4+24*n^3-21*n^2-12*n+12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 12 + 12*x, P(2,x) = 54 + 108*x + 54*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 12 + 12*x;
P[2, x_] := 54 + 108*x + 54*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^6 - 2*n^4 + n^2 + (3*n^6 - 6*n^4 + 3*n^2)*x + (3*n^6 - 6*n^4 + 3*n^2)*x^2 + (n^6 - 2*n^4 + n^2)*x^3)*P[n - 3, x] - (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2 + (6*n^6 - 12*n^5 - 18*n^4 + 24*n^3 + 24*n^2)*x + (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2)*x^2)*P[n - 2, x] + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12 + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12)*x)*P[n - 1, x])/(n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)]];
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] = 12 + 12*x; v[3] = 54 + 108*x + 54*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^6 - 2*n^4 + n^2 + (3*n^6 - 6*n^4 + 3*n^2)*x + (3*n^6 - 6*n^4 + 3*n^2)*x^2 + (n^6 - 2*n^4 + n^2)*x^3)*v[i - 3] - (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2 + (6*n^6 - 12*n^5 - 18*n^4 + 24*n^3 + 24*n^2)*x + (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2)*x^2)*v[i - 2] + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12 + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12)*x)*v[i - 1])/(n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.