Real-rooted atlas

A038208 → Veronese 2 (j=1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12 + 18\,x - 4\,x^{2} + 17\,n - 22\,n x + 4\,n x^{2} - 6\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-24\,x + 56\,x^{2} - 16\,x^{3} + 16\,n - 4\,n x - 36\,n x^{2} + 8\,n x^{3} - 6\,n^{2} - 6\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(16\,x - 56\,x^{2} + 56\,x^{3} - 16\,x^{4} - 12\,n x + 28\,n x^{2} - 16\,n x^{3} + 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
288
332840
412864192
548038432896
6166420483844096
753769856307212818432
8163844300820480204881920
94761617203211980820480512360448
10133120638976626688163840102401572864
1136044822302722973696112640012288020486815744
12950272738918412976128684851211468804915229360128
13244940823429120527155203748659290521606881288192125829120
14619315271565312201490432187432960626196487340032229376536870912
Technical details
Node id
n-368188ccac50ac08 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-efd07601c6f432c5

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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:     8;
  3:    32,      8;
  4:   128,     64;
  5:   480,    384,     32;
  6:  1664,   2048,    384;
  7:  5376,   9856,   3072,   128;
  8: 16384,  43008,  20480,  2048;
  9: 47616, 172032, 119808, 20480, 512;
%F Formula
The row polynomials satisfy (n^3-6*n^2+17*n-12+(4*n^2-22*n+18)*x+(4*n-4)*x^2)*P(n,x) = (2*n^3-6*n^2+16*n+(2*n^3-6*n^2-4*n-24)*x+(8*n^2-36*n+56)*x^2+(8*n-16)*x^3)*P(n-1,x) + ((4*n^2-12*n+16)*x-(4*n^2-28*n+56)*x^2-(16*n-56)*x^3-16*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[((2*n^3 - 6*n^2 + 16*n + (2*n^3 - 6*n^2 - 4*n - 24)*x + (8*n^2 - 36*n + 56)*x^2 + (8*n - 16)*x^3)*P[n - 1, x] + ((4*n^2 - 12*n + 16)*x - (4*n^2 - 28*n + 56)*x^2 - (16*n - 56)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(n^3 - 6*n^2 + 17*n - 12 + (4*n^2 - 22*n + 18)*x + (4*n - 4)*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] = ((2*n^3 - 6*n^2 + 16*n + (2*n^3 - 6*n^2 - 4*n - 24)*x + (8*n^2 - 36*n + 56)*x^2 + (8*n - 16)*x^3)*v[i - 1] + ((4*n^2 - 12*n + 16)*x - (4*n^2 - 28*n + 56)*x^2 - (16*n - 56)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(n^3 - 6*n^2 + 17*n - 12 + (4*n^2 - 22*n + 18)*x + (4*n - 4)*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. [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.