A008292 → γ (shift +1) → D+1

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.

Triangle begins:
  0:    1;
  1:    1;
  2:    3,     2;
  3:    9,     8;
  4:   23,    54,     16;
  5:   53,   324,    136;
  6:  115,  1554,   1536,    272;
  7:  241,  6384,  14976,   3968;
  8:  495, 23726, 114528,  66048,   7936;
  9: 1005, 82556, 731208, 937728, 176896;

The row polynomials satisfy (2*n-5+8*x)*P(n,x) = -(4*n-6+(8*n^2-46*n+67)*x+(8*n^3-64*n^2+190*n-256)*x^2+(48*n^2-264*n+368)*x^3+(64*n-128)*x^4)*P(n-2,x) - ((2*n-3)*x+(4*n^2-24*n+35)*x^2-(16*n^2-88*n+144)*x^3-(96*n-240)*x^4-128*x^5)*d/dx P(n-2,x) + (6*n-13+(8*n^2-46*n+89)*x+(40*n-132)*x^2+32*x^3)*P(n-1,x) + ((2*n-5)*x-(8*n-28)*x^2-32*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[(-(4*n - 6 + (8*n^2 - 46*n + 67)*x + (8*n^3 - 64*n^2 + 190*n - 256)*x^2 + (48*n^2 - 264*n + 368)*x^3 + (64*n - 128)*x^4)*P[n - 2, x] - ((2*n - 3)*x + (4*n^2 - 24*n + 35)*x^2 - (16*n^2 - 88*n + 144)*x^3 - (96*n - 240)*x^4 - 128*x^5)*D[P[n - 2, x], x] + (6*n - 13 + (8*n^2 - 46*n + 89)*x + (40*n - 132)*x^2 + 32*x^3)*P[n - 1, x] + ((2*n - 5)*x - (8*n - 28)*x^2 - 32*x^3)*D[P[n - 1, x], x])/(2*n - 5 + 8*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] = (-(4*n - 6 + (8*n^2 - 46*n + 67)*x + (8*n^3 - 64*n^2 + 190*n - 256)*x^2 + (48*n^2 - 264*n + 368)*x^3 + (64*n - 128)*x^4)*v[i - 2] - ((2*n - 3)*x + (4*n^2 - 24*n + 35)*x^2 - (16*n^2 - 88*n + 144)*x^3 - (96*n - 240)*x^4 - 128*x^5)*deriv(v[i - 2], x) + (6*n - 13 + (8*n^2 - 46*n + 89)*x + (40*n - 132)*x^2 + 32*x^3)*v[i - 1] + ((2*n - 5)*x - (8*n - 28)*x^2 - 32*x^3)*deriv(v[i - 1], x))/(2*n - 5 + 8*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-2394f256e051d5b6
• 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-2394f256e051d5b6
• 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-2394f256e051d5b6

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-2394f256e051d5b6

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

  • premise (sequence): Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 1 e-e49ea542bfed6bbf

      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.