Real-rooted atlas

A034870 → Veronese 2 (j=1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 8\,x - x^{2} + n - 27\,n x + 7\,n x^{2} + 29\,n^{2} x - 19\,n^{2} x^{2} - 10\,n^{3} x + 25\,n^{3} x^{2} - 16\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-16\,x - 28\,x^{2} - 4\,x^{3} + n + 4\,n x - 19\,n x^{2} - 2\,n x^{3} + 19\,n^{2} x + 99\,n^{2} x^{2} + 22\,n^{2} x^{3} - 10\,n^{3} x - 41\,n^{3} x^{2} - 9\,n^{3} x^{3} - 12\,n^{4} x^{2} - 20\,n^{4} x^{3} + 4\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} - 6\,x^{3} - 2\,x^{4} + 8\,n x^{2} - 6\,n x^{3} - 2\,n x^{4} - 20\,n^{2} x^{2} + 10\,n^{2} x^{3} + 10\,n^{2} x^{4} + 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
241620
368054140
48224504128864
510480226819202504928
612880712812672550043226624
714145618018549125005013104686138240

All 15 rows

Technical details
Node id
n-9e973f6dccdaa9ea · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b4db67cf47c9c0a7

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:  2;
  2:  4,   16;
  3:  6,   80,    54;
  4:  8,  224,   504,    128;
  5: 10,  480,  2268,   1920,     250;
  6: 12,  880,  7128,  12672,    5500,     432;
  7: 14, 1456, 18018,  54912,   50050,   13104,    686;
  8: 16, 2240, 39312, 183040,  286000,  157248,  27440,  1024;
  9: 18, 3264, 77112, 509184, 1215500, 1145664, 419832, 52224, 1458;
%F Formula
The row polynomials satisfy (n-1-(10*n^3-29*n^2+27*n-8)*x+(4*n^5-16*n^4+25*n^3-19*n^2+7*n-1)*x^2)*P(n,x) = (n-(10*n^3-19*n^2-4*n+16)*x+(4*n^5-12*n^4-41*n^3+99*n^2-19*n-28)*x^2+(12*n^5-20*n^4-9*n^3+22*n^2-2*n-4)*x^3)*P(n-1,x) + (2*x-(20*n^2-8*n-6)*x^2+(8*n^4+10*n^2-6*n-6)*x^3-(8*n^4-10*n^2+2*n+2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n - (10*n^3 - 19*n^2 - 4*n + 16)*x + (4*n^5 - 12*n^4 - 41*n^3 + 99*n^2 - 19*n - 28)*x^2 + (12*n^5 - 20*n^4 - 9*n^3 + 22*n^2 - 2*n - 4)*x^3)*P[n - 1, x] + (2*x - (20*n^2 - 8*n - 6)*x^2 + (8*n^4 + 10*n^2 - 6*n - 6)*x^3 - (8*n^4 - 10*n^2 + 2*n + 2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (10*n^3 - 29*n^2 + 27*n - 8)*x + (4*n^5 - 16*n^4 + 25*n^3 - 19*n^2 + 7*n - 1)*x^2)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n - (10*n^3 - 19*n^2 - 4*n + 16)*x + (4*n^5 - 12*n^4 - 41*n^3 + 99*n^2 - 19*n - 28)*x^2 + (12*n^5 - 20*n^4 - 9*n^3 + 22*n^2 - 2*n - 4)*x^3)*v[i - 1] + (2*x - (20*n^2 - 8*n - 6)*x^2 + (8*n^4 + 10*n^2 - 6*n - 6)*x^3 - (8*n^4 - 10*n^2 + 2*n + 2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (10*n^3 - 29*n^2 + 27*n - 8)*x + (4*n^5 - 16*n^4 + 25*n^3 - 19*n^2 + 7*n - 1)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.