A008292 → γ (scale 1/2) (shift +1) → D+1

OEIS matches ?

• A034428: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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.

Triangle begins:
  0:   1;
  1:   1;
  2:   2,     1;
  3:   5,     4;
  4:  12,    19,     4;
  5:  27,    94,    34;
  6:  58,   417,   282,    34;
  7: 121,  1656,  2256,   496;
  8: 248,  6055, 15768,  6272,   496;
  9: 503, 20890, 96498, 72992, 11056;

The row polynomials satisfy (n-3+4*x)*P(n,x) = -(2*n-4+(2*n^2-13*n+26)*x+(n^3-9*n^2+34*n-60)*x^2+(6*n^2-36*n+56)*x^3+(8*n-16)*x^4)*P(n-2,x) - ((n-2)*x+(n^2-7*n+14)*x^2-(2*n^2-16*n+36)*x^3-(12*n-40)*x^4-16*x^5)*d/dx P(n-2,x) + (3*n-8+(2*n^2-13*n+33)*x+(10*n-36)*x^2+8*x^3)*P(n-1,x) + ((n-3)*x-(2*n-10)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (2*n^2 - 13*n + 26)*x + (n^3 - 9*n^2 + 34*n - 60)*x^2 + (6*n^2 - 36*n + 56)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] - ((n - 2)*x + (n^2 - 7*n + 14)*x^2 - (2*n^2 - 16*n + 36)*x^3 - (12*n - 40)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (3*n - 8 + (2*n^2 - 13*n + 33)*x + (10*n - 36)*x^2 + 8*x^3)*P[n - 1, x] + ((n - 3)*x - (2*n - 10)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n - 3 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 4 + (2*n^2 - 13*n + 26)*x + (n^3 - 9*n^2 + 34*n - 60)*x^2 + (6*n^2 - 36*n + 56)*x^3 + (8*n - 16)*x^4)*v[i - 2] - ((n - 2)*x + (n^2 - 7*n + 14)*x^2 - (2*n^2 - 16*n + 36)*x^3 - (12*n - 40)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (3*n - 8 + (2*n^2 - 13*n + 33)*x + (10*n - 36)*x^2 + 8*x^3)*v[i - 1] + ((n - 3)*x - (2*n - 10)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(n - 3 + 4*x)); 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)

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-3526da15743474dd
• 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-3526da15743474dd
• 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-3526da15743474dd

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

• preserver_image derivative_polynomial, shift 0 e-3526da15743474dd

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle read by rows: coefficients d(n,k) of André polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1/2, shift 1 e-ea229ed6e1820773

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): eulerian
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-ab3cdcfae289b48d root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • parent_provers root_domain e-6c46c948823f0a8c root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-c61ecf4ef1c45372

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): eulerian
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-ab3cdcfae289b48d (expanded above)
            • ↵ parent_provers root_domain e-6c46c948823f0a8c (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.