Real-rooted atlas

A127773 → θ+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-12\,n + 22\,n x - 12\,n x^{2} + 2\,n x^{3} + 40\,n^{2} - 57\,n^{2} x + 24\,n^{2} x^{2} - 3\,n^{2} x^{3} - 51\,n^{3} + 53\,n^{3} x - 15\,n^{3} x^{2} + n^{3} x^{3} + 31\,n^{4} - 21\,n^{4} x + 3\,n^{4} x^{2} - 9\,n^{5} + 3\,n^{5} x + n^{6}\right)\,P_{n}(x) = \left(-2\,n x^{4} + 3\,n x^{5} - n x^{6} - 2\,n^{2} x^{3} + 6\,n^{2} x^{4} - 3\,n^{2} x^{5} + 3\,n^{3} x^{3} - n^{3} x^{4} - 3\,n^{3} x^{5} + n^{3} x^{6} + n^{4} x^{3} - 6\,n^{4} x^{4} + 3\,n^{4} x^{5} - 3\,n^{5} x^{3} + 3\,n^{5} x^{4} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(18\,n x^{3} - 24\,n x^{4} + 6\,n x^{5} + 18\,n^{2} x^{2} - 39\,n^{2} x^{3} + 6\,n^{2} x^{4} + 3\,n^{2} x^{5} - 15\,n^{3} x^{2} - 15\,n^{3} x^{3} + 21\,n^{3} x^{4} - 3\,n^{3} x^{5} - 15\,n^{4} x^{2} + 33\,n^{4} x^{3} - 9\,n^{4} x^{4} + 15\,n^{5} x^{2} - 9\,n^{5} x^{3} - 3\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(36\,x^{2} - 30\,x^{3} + 6\,x^{4} + 36\,n x - 78\,n x^{2} + 33\,n x^{3} - 3\,n x^{4} - 48\,n^{2} x + 12\,n^{2} x^{2} + 21\,n^{2} x^{3} - 6\,n^{2} x^{4} - 15\,n^{3} x + 69\,n^{3} x^{2} - 33\,n^{3} x^{3} + 3\,n^{3} x^{4} + 45\,n^{4} x - 48\,n^{4} x^{2} + 9\,n^{4} x^{3} - 21\,n^{5} x + 9\,n^{5} x^{2} + 3\,n^{6} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 6\,x\)
\(P_{2}(x) = 36\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16612
20361854
30012040160
400030075375
50000630126756
60000011761961372
700000020162882304
8000000032404053645
90000000049505505500
1000000000072607267986
1100000000001029693611232
120000000000014196118315379
1300000000000019110147020580
14000000000000025200180027000
Technical details
Node id
n-c055169f8364a43f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2327141074f67bdf

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 6,  6;
  2: 0, 36,  18;
  3: 0,  0, 120,  40;
  4: 0,  0,   0, 300,  75;
  5: 0,  0,   0,   0, 630,  126;
  6: 0,  0,   0,   0,   0, 1176,  196;
  7: 0,  0,   0,   0,   0,    0, 2016,  288;
  8: 0,  0,   0,   0,   0,    0,    0, 3240,  405;
  9: 0,  0,   0,   0,   0,    0,    0,    0, 4950, 550;
%F Formula
The row polynomials satisfy (n^6-9*n^5+31*n^4-51*n^3+40*n^2-12*n+(3*n^5-21*n^4+53*n^3-57*n^2+22*n)*x+(3*n^4-15*n^3+24*n^2-12*n)*x^2+(n^3-3*n^2+2*n)*x^3)*P(n,x) = ((n^6-3*n^5+n^4+3*n^3-2*n^2)*x^3+(3*n^5-6*n^4-n^3+6*n^2-2*n)*x^4+(3*n^4-3*n^3-3*n^2+3*n)*x^5+(n^3-n)*x^6)*P(n-3,x) - ((3*n^6-15*n^5+15*n^4+15*n^3-18*n^2)*x^2+(9*n^5-33*n^4+15*n^3+39*n^2-18*n)*x^3+(9*n^4-21*n^3-6*n^2+24*n)*x^4+(3*n^3-3*n^2-6*n)*x^5)*P(n-2,x) + ((3*n^6-21*n^5+45*n^4-15*n^3-48*n^2+36*n)*x+(9*n^5-48*n^4+69*n^3+12*n^2-78*n+36)*x^2+(9*n^4-33*n^3+21*n^2+33*n-30)*x^3+(3*n^3-6*n^2-3*n+6)*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 6*x, P(2,x) = 36*x + 18*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[2, x_] := 36*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n^6 - 3*n^5 + n^4 + 3*n^3 - 2*n^2)*x^3 + (3*n^5 - 6*n^4 - n^3 + 6*n^2 - 2*n)*x^4 + (3*n^4 - 3*n^3 - 3*n^2 + 3*n)*x^5 + (n^3 - n)*x^6)*P[n - 3, x] - ((3*n^6 - 15*n^5 + 15*n^4 + 15*n^3 - 18*n^2)*x^2 + (9*n^5 - 33*n^4 + 15*n^3 + 39*n^2 - 18*n)*x^3 + (9*n^4 - 21*n^3 - 6*n^2 + 24*n)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*P[n - 2, x] + ((3*n^6 - 21*n^5 + 45*n^4 - 15*n^3 - 48*n^2 + 36*n)*x + (9*n^5 - 48*n^4 + 69*n^3 + 12*n^2 - 78*n + 36)*x^2 + (9*n^4 - 33*n^3 + 21*n^2 + 33*n - 30)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*P[n - 1, x])/(n^6 - 9*n^5 + 31*n^4 - 51*n^3 + 40*n^2 - 12*n + (3*n^5 - 21*n^4 + 53*n^3 - 57*n^2 + 22*n)*x + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)]];
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] = 6 + 6*x; v[3] = 36*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = (((n^6 - 3*n^5 + n^4 + 3*n^3 - 2*n^2)*x^3 + (3*n^5 - 6*n^4 - n^3 + 6*n^2 - 2*n)*x^4 + (3*n^4 - 3*n^3 - 3*n^2 + 3*n)*x^5 + (n^3 - n)*x^6)*v[i - 3] - ((3*n^6 - 15*n^5 + 15*n^4 + 15*n^3 - 18*n^2)*x^2 + (9*n^5 - 33*n^4 + 15*n^3 + 39*n^2 - 18*n)*x^3 + (9*n^4 - 21*n^3 - 6*n^2 + 24*n)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*v[i - 2] + ((3*n^6 - 21*n^5 + 45*n^4 - 15*n^3 - 48*n^2 + 36*n)*x + (9*n^5 - 48*n^4 + 69*n^3 + 12*n^2 - 78*n + 36)*x^2 + (9*n^4 - 33*n^3 + 21*n^2 + 33*n - 30)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*v[i - 1])/(n^6 - 9*n^5 + 31*n^4 - 51*n^3 + 40*n^2 - 12*n + (3*n^5 - 21*n^4 + 53*n^3 - 57*n^2 + 22*n)*x + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)); 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.