Real-rooted atlas

A034870 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 9\,x - 2\,x^{2} + 9\,n - 13\,n x + 2\,n x^{2} - 7\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(-1 + 12\,x + 19\,x^{2} - 6\,x^{3} + 2\,n - 5\,n x - 31\,n x^{2} + 6\,n x^{3} - 3\,n^{2} - 13\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{3} + 6\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x - 8\,x^{2} + 14\,x^{3} - 4\,x^{4} - 4\,n x + 12\,n x^{2} - 8\,n x^{3} + 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
278116
3164518180
429168154321384
5464658403905011792
6671056326729048257218192
792209310010150158008154798136864
812137602584459488529101892826601281163840
91546273587521935962625481551423998442841621720896
101919880121125542640104975094057639729077520655520013145728
112321486123115413536933553000452652228850369166741404489625242113631488
122772152841441430764801054175118186192164329487845024194683524085613662288158720256
1332630225705640647933028120950632579357726160052151580193722103864575394680188503381251658240
1437941328115069512809160687236551956536403112379552834906401485297454420416072522456224402538939211073741824
Technical details
Node id
n-12af8335457074fa · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4af00dfa4d313d82

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,    1;
  2:   7,    8,     1;
  3:  16,   45,    18,      1;
  4:  29,  168,   154,     32,      1;
  5:  46,  465,   840,    390,     50,      1;
  6:  67, 1056,  3267,   2904,    825,     72,     1;
  7:  92, 2093, 10010,  15015,   8008,   1547,    98,    1;
  8: 121, 3760, 25844,  59488,  52910,  18928,  2660,  128,   1;
  9: 154, 6273, 58752, 193596, 262548, 155142, 39984, 4284, 162, 1;
%F Formula
The row polynomials satisfy (2*n^3-7*n^2+9*n-4+(4*n^2-13*n+9)*x+(2*n-2)*x^2)*P(n,x) = (2*n^3-3*n^2+2*n-1+(6*n^3-13*n^2-5*n+12)*x+(12*n^2-31*n+19)*x^2+(6*n-6)*x^3)*P(n-1,x) + ((4*n^2-4*n-2)*x-(4*n^2-12*n+8)*x^2-(8*n-14)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 3*n^2 + 2*n - 1 + (6*n^3 - 13*n^2 - 5*n + 12)*x + (12*n^2 - 31*n + 19)*x^2 + (6*n - 6)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n - 2)*x - (4*n^2 - 12*n + 8)*x^2 - (8*n - 14)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 7*n^2 + 9*n - 4 + (4*n^2 - 13*n + 9)*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] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^3 - 3*n^2 + 2*n - 1 + (6*n^3 - 13*n^2 - 5*n + 12)*x + (12*n^2 - 31*n + 19)*x^2 + (6*n - 6)*x^3)*v[i - 1] + ((4*n^2 - 4*n - 2)*x - (4*n^2 - 12*n + 8)*x^2 - (8*n - 14)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 7*n^2 + 9*n - 4 + (4*n^2 - 13*n + 9)*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.