Real-rooted atlas

A013610 → Veronese 2 (j=1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(-60 + 130\,x - 36\,x^{2} + 105\,n - 166\,n x + 36\,n x^{2} - 54\,n^{2} + 36\,n^{2} x + 9\,n^{3}\right)\,P_{n}(x) = \left(-684\,x + 1692\,x^{2} - 648\,x^{3} + 24\,n + 1022\,n x - 1458\,n x^{2} + 324\,n x^{3} - 27\,n^{2} - 531\,n^{2} x + 324\,n^{2} x^{2} + 9\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(40\,x - 476\,x^{2} + 1116\,x^{3} - 648\,x^{4} - 54\,n x + 558\,n x^{2} - 648\,n x^{3} + 18\,n^{2} x - 162\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
266
3362763
4120108228
52857562431284
6558345614585472
79661115111664218725968
81536287286609617496113856
922956350426681415746419683509760
10327012571284855610497601968302224128
1144882289872277396505197019683001771479708288
12597639085253887681905314414959080212576441923584
137761633204115714176004627283160675233834041594323180397056
1498709828002300383816512724836181290619840464022320522771661824
Technical details
Node id
n-9bf2a8283c6b49b5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3391db1fdb20fe14

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:    3;
  2:    6;
  3:   36,    27;
  4:  120,   108;
  5:  285,   756,    243;
  6:  558,  3456,   1458;
  7:  966, 11151,  11664,   2187;
  8: 1536, 28728,  66096,  17496;
  9: 2295, 63504, 266814, 157464, 19683;
%F Formula
The row polynomials satisfy (9*n^3-54*n^2+105*n-60+(36*n^2-166*n+130)*x+(36*n-36)*x^2)*P(n,x) = (9*n^3-27*n^2+24*n+(81*n^3-531*n^2+1022*n-684)*x+(324*n^2-1458*n+1692)*x^2+(324*n-648)*x^3)*P(n-1,x) + ((18*n^2-54*n+40)*x-(162*n^2-558*n+476)*x^2-(648*n-1116)*x^3-648*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 27*n^2 + 24*n + (81*n^3 - 531*n^2 + 1022*n - 684)*x + (324*n^2 - 1458*n + 1692)*x^2 + (324*n - 648)*x^3)*P[n - 1, x] + ((18*n^2 - 54*n + 40)*x - (162*n^2 - 558*n + 476)*x^2 - (648*n - 1116)*x^3 - 648*x^4)*D[P[n - 1, x], x])/(9*n^3 - 54*n^2 + 105*n - 60 + (36*n^2 - 166*n + 130)*x + (36*n - 36)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 27*n^2 + 24*n + (81*n^3 - 531*n^2 + 1022*n - 684)*x + (324*n^2 - 1458*n + 1692)*x^2 + (324*n - 648)*x^3)*v[i - 1] + ((18*n^2 - 54*n + 40)*x - (162*n^2 - 558*n + 476)*x^2 - (648*n - 1116)*x^3 - 648*x^4)*deriv(v[i - 1], x))/(9*n^3 - 54*n^2 + 105*n - 60 + (36*n^2 - 166*n + 130)*x + (36*n - 36)*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

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.