Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 2\,x + n\right)\,P_{n}(x) = \left(12 + 22\,x + 20\,x^{2} + 10\,x^{3} + 2\,x^{4} + 3\,n - 4\,n x - 13\,n x^{2} - 9\,n x^{3} - 2\,n x^{4} - 16\,n^{2} - 26\,n^{2} x - 15\,n^{2} x^{2} - 3\,n^{2} x^{3} - 4\,n^{3} - 4\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(48 + 120\,x + 124\,x^{2} + 66\,x^{3} + 18\,x^{4} + 2\,x^{5} + 44\,n + 88\,n x + 67\,n x^{2} + 23\,n x^{3} + 3\,n x^{4} + 8\,n^{2} + 12\,n^{2} x + 6\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(3 + 6\,x + 6\,x^{2} + 2\,x^{3} + 14\,n + 16\,n x + 5\,n x^{2} + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-12 - 20\,x - 11\,x^{2} - 2\,x^{3} - 4\,n - 4\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
2189128
39572161184
45796002082711415
539665285256354246112403
629944491333105894021366811120985
724614148188037835615122232131342014811293299

All 15 rows

Technical details
Node id
n-e0819b503f75bf84 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fbd847c84279d5f6

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:        4,        1;
  2:       18,        9,        1;
  3:       95,       72,       16,        1;
  4:      579,      600,      208,       27,        1;
  5:     3966,     5285,     2563,      542,       46,       1;
  6:    29944,    49133,    31058,     9402,     1366,      81,      1;
  7:   246141,   481880,   378356,   151222,    32131,    3420,    148,     1;
  8:  2183173,  4981560,  4696852,  2352330,   668361,  106180,   8604,   279,   1;
  9: 20744088, 54181953, 59837710, 36224700, 13015374, 2802223, 346179, 21882, 538, 1;
%F Formula
The row polynomials satisfy (n+3+2*x)*P(n,x) = -(4*n^3+16*n^2-3*n-12+(4*n^3+26*n^2+4*n-22)*x+(n^3+15*n^2+13*n-20)*x^2+(3*n^2+9*n-10)*x^3+(2*n-2)*x^4)*P(n-2,x) + (8*n^2+44*n+48+(12*n^2+88*n+120)*x+(6*n^2+67*n+124)*x^2+(n^2+23*n+66)*x^3+(3*n+18)*x^4+2*x^5)*d/dx P(n-2,x) + (4*n^2+14*n+3+(2*n^2+16*n+6)*x+(5*n+6)*x^2+2*x^3)*P(n-1,x) - (4*n+12+(4*n+20)*x+(n+11)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 + 16*n^2 - 3*n - 12 + (4*n^3 + 26*n^2 + 4*n - 22)*x + (n^3 + 15*n^2 + 13*n - 20)*x^2 + (3*n^2 + 9*n - 10)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + (8*n^2 + 44*n + 48 + (12*n^2 + 88*n + 120)*x + (6*n^2 + 67*n + 124)*x^2 + (n^2 + 23*n + 66)*x^3 + (3*n + 18)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (4*n^2 + 14*n + 3 + (2*n^2 + 16*n + 6)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] - (4*n + 12 + (4*n + 20)*x + (n + 11)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 3 + 2*x)]];
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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 + 16*n^2 - 3*n - 12 + (4*n^3 + 26*n^2 + 4*n - 22)*x + (n^3 + 15*n^2 + 13*n - 20)*x^2 + (3*n^2 + 9*n - 10)*x^3 + (2*n - 2)*x^4)*v[i - 2] + (8*n^2 + 44*n + 48 + (12*n^2 + 88*n + 120)*x + (6*n^2 + 67*n + 124)*x^2 + (n^2 + 23*n + 66)*x^3 + (3*n + 18)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (4*n^2 + 14*n + 3 + (2*n^2 + 16*n + 6)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] - (4*n + 12 + (4*n + 20)*x + (n + 11)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 3 + 2*x)); 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. [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.