Real-rooted atlas

A034870 → Veronese 2 (j=1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(-9 + 11\,x - 2\,x^{2} + 16\,n - 15\,n x + 2\,n x^{2} - 9\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(-16\,x + 36\,x^{2} - 8\,x^{3} + 4\,n + 23\,n x - 41\,n x^{2} + 6\,n x^{3} - 3\,n^{2} - 21\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{3} + 6\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x - 20\,x^{2} + 18\,x^{3} - 4\,x^{4} - 8\,n x + 16\,n x^{2} - 8\,n x^{3} + 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
28412
32632664
464168808320
5130624612160101536
6232180431681672280127168
73784368122981144038224481432768
85769296386885720033280772867216147456
983417952104040226304207740832321428096018655360
101160321482480647493601007760633080186048246241320202883584
1115625420853796621602244024580368995216912283812164019417602212582912
122048870321080816557612013788224174730081164867240763367286406274422882454525952
132626134160203918013156000417553507012976064483720327225609044750131560094380291226234881024
143304199836365040028811640114277800246127140299537280209235600836362801874730022651201375923640281006632960
Technical details
Node id
n-7036cf2590641970 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-700e7903acaa52c2

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:   8,     4;
  3:  26,    32,      6;
  4:  64,   168,     80,      8;
  5: 130,   624,    612,    160,     10;
  6: 232,  1804,   3168,   1672,    280,    12;
  7: 378,  4368,  12298,  11440,   3822,   448,    14;
  8: 576,  9296,  38688,  57200,  33280,  7728,   672,  16;
  9: 834, 17952, 104040, 226304, 207740, 83232, 14280, 960, 18;
%F Formula
The row polynomials satisfy (2*n^3-9*n^2+16*n-9+(4*n^2-15*n+11)*x+(2*n-2)*x^2)*P(n,x) = (2*n^3-3*n^2+4*n+(6*n^3-21*n^2+23*n-16)*x+(12*n^2-41*n+36)*x^2+(6*n-8)*x^3)*P(n-1,x) + ((4*n^2-8*n+6)*x-(4*n^2-16*n+20)*x^2-(8*n-18)*x^3-4*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 - 3*n^2 + 4*n + (6*n^3 - 21*n^2 + 23*n - 16)*x + (12*n^2 - 41*n + 36)*x^2 + (6*n - 8)*x^3)*P[n - 1, x] + ((4*n^2 - 8*n + 6)*x - (4*n^2 - 16*n + 20)*x^2 - (8*n - 18)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 16*n - 9 + (4*n^2 - 15*n + 11)*x + (2*n - 2)*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 - 3*n^2 + 4*n + (6*n^3 - 21*n^2 + 23*n - 16)*x + (12*n^2 - 41*n + 36)*x^2 + (6*n - 8)*x^3)*v[i - 1] + ((4*n^2 - 8*n + 6)*x - (4*n^2 - 16*n + 20)*x^2 - (8*n - 18)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 16*n - 9 + (4*n^2 - 15*n + 11)*x + (2*n - 2)*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.