Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → D

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(-12\,n - 14\,n^{2} + 12\,n^{3} + 7\,n^{4} - 6\,n^{5} + n^{6}\right)\,P_{n}(x) = \left(24 + 72\,x + 72\,x^{2} + 24\,x^{3} - 8\,n^{2} - 24\,n^{2} x - 24\,n^{2} x^{2} - 8\,n^{2} x^{3} + 15\,n^{3} + 45\,n^{3} x + 45\,n^{3} x^{2} + 15\,n^{3} x^{3} - 5\,n^{4} - 15\,n^{4} x - 15\,n^{4} x^{2} - 5\,n^{4} x^{3} - 3\,n^{5} - 9\,n^{5} x - 9\,n^{5} x^{2} - 3\,n^{5} x^{3} + n^{6} + 3\,n^{6} x + 3\,n^{6} x^{2} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(12 + 48\,x + 72\,x^{2} + 48\,x^{3} + 12\,x^{4} + 6\,n + 24\,n x + 36\,n x^{2} + 24\,n x^{3} + 6\,n x^{4} - 6\,n^{2} - 24\,n^{2} x - 36\,n^{2} x^{2} - 24\,n^{2} x^{3} - 6\,n^{2} x^{4}\right)\,P_{n-3}'(x) + \left(36\,n + 72\,n x + 36\,n x^{2} + 6\,n^{2} + 12\,n^{2} x + 6\,n^{2} x^{2} - 60\,n^{3} - 120\,n^{3} x - 60\,n^{3} x^{2} + 9\,n^{4} + 18\,n^{4} x + 9\,n^{4} x^{2} + 12\,n^{5} + 24\,n^{5} x + 12\,n^{5} x^{2} - 3\,n^{6} - 6\,n^{6} x - 3\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(-24 - 72\,x - 72\,x^{2} - 24\,x^{3} - 24\,n - 72\,n x - 72\,n x^{2} - 24\,n x^{3} + 12\,n^{2} + 36\,n^{2} x + 36\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-72\,n - 72\,n x - 30\,n^{2} - 30\,n^{2} x + 63\,n^{3} + 63\,n^{3} x + 3\,n^{4} + 3\,n^{4} x - 15\,n^{5} - 15\,n^{5} x + 3\,n^{6} + 3\,n^{6} x\right)\,P_{n-1}(x) + \left(18\,n + 36\,n x + 18\,n x^{2} - 6\,n^{2} - 12\,n^{2} x - 6\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 8 + 6\,x\)
\(P_{3}(x) = 18 + 36\,x + 12\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
28614
318361266
4321089620256
55024036020030880
672450960900360422784
798756210028001890588568288

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-a3755ba558645334 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-18d84775e29cd07e

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   2;
  2:   8,    6;
  3:  18,   36,   12;
  4:  32,  108,   96,    20;
  5:  50,  240,  360,   200,    30;
  6:  72,  450,  960,   900,   360,    42;
  7:  98,  756, 2100,  2800,  1890,   588,   56;
  8: 128, 1176, 4032,  7000,  6720,  3528,  896,   72;
  9: 162, 1728, 7056, 15120, 18900, 14112, 6048, 1296, 90;
%F Formula
The row polynomials satisfy (n^6-6*n^5+7*n^4+12*n^3-14*n^2-12*n)*P(n,x) = (n^6-3*n^5-5*n^4+15*n^3-8*n^2+24+(3*n^6-9*n^5-15*n^4+45*n^3-24*n^2+72)*x+(3*n^6-9*n^5-15*n^4+45*n^3-24*n^2+72)*x^2+(n^6-3*n^5-5*n^4+15*n^3-8*n^2+24)*x^3)*P(n-3,x) - (6*n^2-6*n-12+(24*n^2-24*n-48)*x+(36*n^2-36*n-72)*x^2+(24*n^2-24*n-48)*x^3+(6*n^2-6*n-12)*x^4)*d/dx P(n-3,x) - (3*n^6-12*n^5-9*n^4+60*n^3-6*n^2-36*n+(6*n^6-24*n^5-18*n^4+120*n^3-12*n^2-72*n)*x+(3*n^6-12*n^5-9*n^4+60*n^3-6*n^2-36*n)*x^2)*P(n-2,x) + (12*n^2-24*n-24+(36*n^2-72*n-72)*x+(36*n^2-72*n-72)*x^2+(12*n^2-24*n-24)*x^3)*d/dx P(n-2,x) + (3*n^6-15*n^5+3*n^4+63*n^3-30*n^2-72*n+(3*n^6-15*n^5+3*n^4+63*n^3-30*n^2-72*n)*x)*P(n-1,x) - (6*n^2-18*n+(12*n^2-36*n)*x+(6*n^2-18*n)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 8 + 6*x, P(3,x) = 18 + 36*x + 12*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 8 + 6*x;
P[3, x_] := 18 + 36*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24)*x^3)*P[n - 3, x] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*D[P[n - 3, x], x] - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 12*n^2 - 72*n)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n)*x^2)*P[n - 2, x] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*D[P[n - 2, x], x] + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n)*x)*P[n - 1, x] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*D[P[n - 1, x], x])/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)]];
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; v[3] = 8 + 6*x; v[4] = 18 + 36*x + 12*x^2; for(n = 4, N, my(i = n + 1); v[i] = ((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24)*x^3)*v[i - 3] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*deriv(v[i - 3], x) - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 12*n^2 - 72*n)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n)*x^2)*v[i - 2] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*deriv(v[i - 2], x) + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n)*x)*v[i - 1] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*deriv(v[i - 1], x))/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-98dce694e357661a
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-98dce694e357661a
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-98dce694e357661a

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.