Real-rooted atlas

A048993 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-x^{2} + x^{3}\right)\,P_{n-3}'(x) + \left(1 + x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(3\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 2 + 8\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
228313
322021447
424493445188
5292345300806825
621881173166077513273937
723803801820459501722203820270
827641197337884400051747234302969111835
9215323706516750024738015283844394628841410657423
10230681134937194201446275121388448906910104010800560114097622
112614034508130290048130650901883448396811367256210870176007371226965867
122122841044453125761244443950563869784443997401646911234359934103002746794813186685725
13224572315178551697100237917680436546110385312928182495456494516887920770753467413401196141355314108
14249148949221321099078012542097752904777876320649429119003212806503725801341540201700098413180446033314841510289242825
Technical details
Node id
n-23b51df229a9c1bc · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-91c77fcab8e661c4

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: 2,    2;
  2: 2,    8,     3;
  3: 2,   20,    21,      4;
  4: 2,   44,    93,     44,      5;
  5: 2,   92,   345,    300,     80,      6;
  6: 2,  188,  1173,   1660,    775,    132,     7;
  7: 2,  380,  3801,   8204,   5950,   1722,   203,    8;
  8: 2,  764, 11973,  37884,  40005,  17472,  3430,  296,   9;
  9: 2, 1532, 37065, 167500, 247380, 152838, 44394, 6288, 414, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(x^2-x^3)*P(n-3,x) - (x^2-x^3)*d/dx P(n-3,x) + (1+x-3*x^2)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + 3*x*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 2 + 8*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 2 + 8*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(x^2 - x^3)*P[n - 3, x] - (x^2 - x^3)*D[P[n - 3, x], x] + (1 + x - 3*x^2)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + 3*x*P[n - 1, x] + x*D[P[n - 1, x], 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] = 2 + 2*x; v[3] = 2 + 8*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = -(x^2 - x^3)*v[i - 3] - (x^2 - x^3)*deriv(v[i - 3], x) + (1 + x - 3*x^2)*v[i - 2] + (x - 2*x^2)*deriv(v[i - 2], x) + 3*x*v[i - 1] + x*deriv(v[i - 1], 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.