Real-rooted atlas

A003506 → (θ+1)² → D+1

nonnegative coefficients

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

Recurrence

\(\left(7 - 11\,x + 5\,x^{2} - x^{3} - 11\,n + 25\,n x - 11\,n x^{2} + 3\,n x^{3} + 4\,n^{2} - 19\,n^{2} x + 9\,n^{2} x^{2} - 3\,n^{2} x^{3} + 5\,n^{3} x - 4\,n^{3} x^{2} + n^{3} x^{3} + n^{4} x^{2}\right)\,P_{n}(x) = \left(-3\,n - n x + n x^{2} - n x^{3} + 4\,n^{2} - n^{2} x^{2} + 3\,n^{2} x^{3} + 5\,n^{3} x + 5\,n^{3} x^{2} + n^{3} x^{3} + n^{3} x^{4} + n^{4} x^{2} + n^{4} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
210818
3277827132
45226430064680
58562012308201252880
6126120034203960183021610752
7175205876651274010185357034336736

All 15 rows

Technical details
Node id
n-97198ddb2970addc · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-cb516ecd7bb869fb

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:   0;
  1:   1;
  2:  10,    8;
  3:  27,   78,    27;
  4:  52,  264,   300,    64;
  5:  85,  620,  1230,   820,    125;
  6: 126, 1200,  3420,  3960,   1830,   216;
  7: 175, 2058,  7665, 12740,  10185,  3570,   343;
  8: 232, 3248, 14952, 32480,  37240, 22512,  6328,   512;
  9: 297, 4824, 26460, 71064, 106470, 92232, 44604, 10440, 729;
%F Formula
The row polynomials satisfy (4*n^2-11*n+7+(5*n^3-19*n^2+25*n-11)*x+(n^4-4*n^3+9*n^2-11*n+5)*x^2+(n^3-3*n^2+3*n-1)*x^3)*P(n,x) = (4*n^2-3*n+(5*n^3-n)*x+(n^4+5*n^3-n^2+n)*x^2+(n^4+n^3+3*n^2-n)*x^3+n^3*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 3*n + (5*n^3 - n)*x + (n^4 + 5*n^3 - n^2 + n)*x^2 + (n^4 + n^3 + 3*n^2 - n)*x^3 + n^3*x^4)*P[n - 1, x])/(4*n^2 - 11*n + 7 + (5*n^3 - 19*n^2 + 25*n - 11)*x + (n^4 - 4*n^3 + 9*n^2 - 11*n + 5)*x^2 + (n^3 - 3*n^2 + 3*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 3*n + (5*n^3 - n)*x + (n^4 + 5*n^3 - n^2 + n)*x^2 + (n^4 + n^3 + 3*n^2 - n)*x^3 + n^3*x^4)*v[i - 1])/(4*n^2 - 11*n + 7 + (5*n^3 - 19*n^2 + 25*n - 11)*x + (n^4 - 4*n^3 + 9*n^2 - 11*n + 5)*x^2 + (n^3 - 3*n^2 + 3*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.