Real-rooted atlas

A126351 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-x + x^{3} + n x^{3}\right)\,P_{n}(x) = \left(2 + 3\,x - 4\,x^{2} - 6\,x^{3} + 2\,n + 7\,n x - 16\,n x^{3} - 12\,n x^{4} - 2\,n x^{5} + n^{2} x - 8\,n^{2} x^{3} - 10\,n^{2} x^{4} - 3\,n^{2} x^{5} - n^{3} x^{3} - 2\,n^{3} x^{4} - n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-1 - 5\,x - 6\,x^{2} + 5\,x^{3} + 15\,x^{4} + 10\,x^{5} + 2\,x^{6} - n x - 2\,n x^{2} + 4\,n x^{3} + 13\,n x^{4} + 11\,n x^{5} + 3\,n x^{6} + n^{2} x^{3} + 3\,n^{2} x^{4} + 3\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-2 - 5\,x + 2\,x^{2} + 8\,x^{3} + 2\,x^{4} - 2\,n x + 8\,n x^{3} + 4\,n x^{4} + 2\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} - 2\,x^{4} - x^{5} - n x^{3} - 2\,n x^{4} - n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
210261248
33714212932340
41517661038516802551
5674425876985796185519220473
63263247205607356976281806294448175954
7170071504844117295293363580551266662068510241614986

All 15 rows

Technical details
Node id
n-a08dcc8801b947d3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-fda8f9e2aa8fd28c

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:      3,       4;
  2:     10,      26,       12;
  3:     37,     142,      129,       32;
  4:    151,     766,     1038,      516,       80;
  5:    674,    4258,     7698,     5796,     1855,      192;
  6:   3263,   24720,    56073,    56976,    28180,     6294,      448;
  7:  17007,  150484,   411729,   529336,   358055,   126666,    20685,    1024;
  8:  94828,  961026,  3085176,  4814372,  4158355,  2041950,   543466,   66824,   2304;
  9: 562595, 6432226, 23737746, 43673668, 46092695, 29187138, 10958612, 2267560, 214011, 5120;
%F Formula
The row polynomials satisfy -(x-(n+1)*x^3)*P(n,x) = (2*n+2+(n^2+7*n+3)*x-4*x^2-(n^3+8*n^2+16*n+6)*x^3-(2*n^3+10*n^2+12*n)*x^4-(n^3+3*n^2+2*n)*x^5)*P(n-2,x) - (1+(n+5)*x+(2*n+6)*x^2-(n^2+4*n+5)*x^3-(3*n^2+13*n+15)*x^4-(3*n^2+11*n+10)*x^5-(n^2+3*n+2)*x^6)*d/dx P(n-2,x) - (2+(2*n+5)*x-2*x^2-(2*n^2+8*n+8)*x^3-(2*n^2+4*n+2)*x^4)*P(n-1,x) + (x+2*x^2-n*x^3-(2*n+2)*x^4-(n+1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (n^2 + 7*n + 3)*x - 4*x^2 - (n^3 + 8*n^2 + 16*n + 6)*x^3 - (2*n^3 + 10*n^2 + 12*n)*x^4 - (n^3 + 3*n^2 + 2*n)*x^5)*P[n - 2, x] - (1 + (n + 5)*x + (2*n + 6)*x^2 - (n^2 + 4*n + 5)*x^3 - (3*n^2 + 13*n + 15)*x^4 - (3*n^2 + 11*n + 10)*x^5 - (n^2 + 3*n + 2)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 5)*x - 2*x^2 - (2*n^2 + 8*n + 8)*x^3 - (2*n^2 + 4*n + 2)*x^4)*P[n - 1, x] + (x + 2*x^2 - n*x^3 - (2*n + 2)*x^4 - (n + 1)*x^5)*D[P[n - 1, x], x])/(-x + (n + 1)*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 2 + (n^2 + 7*n + 3)*x - 4*x^2 - (n^3 + 8*n^2 + 16*n + 6)*x^3 - (2*n^3 + 10*n^2 + 12*n)*x^4 - (n^3 + 3*n^2 + 2*n)*x^5)*v[i - 2] - (1 + (n + 5)*x + (2*n + 6)*x^2 - (n^2 + 4*n + 5)*x^3 - (3*n^2 + 13*n + 15)*x^4 - (3*n^2 + 11*n + 10)*x^5 - (n^2 + 3*n + 2)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 5)*x - 2*x^2 - (2*n^2 + 8*n + 8)*x^3 - (2*n^2 + 4*n + 2)*x^4)*v[i - 1] + (x + 2*x^2 - n*x^3 - (2*n + 2)*x^4 - (n + 1)*x^5)*deriv(v[i - 1], x))/(-x + (n + 1)*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

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.