A008292 → γ (scale 1/4) (shift +1) → D+1

OEIS matches ?

• A002301: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 3 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:      2;
  2:      6,       2;
  3:     24,      16;
  4:    104,     120,      16;
  5:    448,     960,     272;
  6:   1888,    7584,    3696,     272;
  7:   7808,   56832,   48384,    7936;
  8:  31872,  403328,  597504,  168960,   7936;
  9: 129024, 2738176, 6828288, 3256320, 353792;

The row polynomials satisfy (n-4+4*x)*P(n,x) = -(8*n-24+(4*n^2-32*n+92)*x+(n^3-11*n^2+56*n-128)*x^2+(6*n^2-42*n+76)*x^3+(8*n-16)*x^4)*P(n-2,x) - ((4*n-12)*x+(2*n^2-18*n+52)*x^2-(2*n^2-26*n+84)*x^3-(12*n-60)*x^4-16*x^5)*d/dx P(n-2,x) + (6*n-22+(2*n^2-16*n+56)*x+(10*n-42)*x^2+8*x^3)*P(n-1,x) + ((2*n-8)*x-(2*n-16)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (4*n^2 - 32*n + 92)*x + (n^3 - 11*n^2 + 56*n - 128)*x^2 + (6*n^2 - 42*n + 76)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] - ((4*n - 12)*x + (2*n^2 - 18*n + 52)*x^2 - (2*n^2 - 26*n + 84)*x^3 - (12*n - 60)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (6*n - 22 + (2*n^2 - 16*n + 56)*x + (10*n - 42)*x^2 + 8*x^3)*P[n - 1, x] + ((2*n - 8)*x - (2*n - 16)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n - 4 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n - 24 + (4*n^2 - 32*n + 92)*x + (n^3 - 11*n^2 + 56*n - 128)*x^2 + (6*n^2 - 42*n + 76)*x^3 + (8*n - 16)*x^4)*v[i - 2] - ((4*n - 12)*x + (2*n^2 - 18*n + 52)*x^2 - (2*n^2 - 26*n + 84)*x^3 - (12*n - 60)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (6*n - 22 + (2*n^2 - 16*n + 56)*x + (10*n - 42)*x^2 + 8*x^3)*v[i - 1] + ((2*n - 8)*x - (2*n - 16)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(n - 4 + 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-ea99ed3c2ecebb89
• 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-ea99ed3c2ecebb89
• 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-ea99ed3c2ecebb89

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-ea99ed3c2ecebb89

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

  • premise (sequence): Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1/4, shift 1 e-00ceaf3777abe095

      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 ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A008292 → γ (scale 1/4) → D+1

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.