A081582 → θ+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;
  2: 1,  18,    3;
  3: 1,  34,   51,     4;
  4: 1,  50,  291,   100,      5;
  5: 1,  66,  723,   964,    165,      6;
  6: 1,  82, 1347,  4644,   2245,    246,     7;
  7: 1,  98, 2163, 13188,  16485,   4326,   343,     8;
  8: 1, 114, 3171, 28644,  73605,  42966,  7399,   456,   9;
  9: 1, 130, 4371, 53060, 224805, 269766, 92855, 11656, 585, 10;

The row polynomials satisfy -(n+(8*n^2-2*n)*x-(8*n^2+15*n)*x^2)*P(n,x) = -(n+(8*n^2+30*n-30)*x+(120*n^2+97*n-228)*x^2-(16*n+30)*x^3)*P(n-1,x) + (x+(8*n+45)*x^2+(240*n+451)*x^3+(8*n+15)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (8*n^2 + 30*n - 30)*x + (120*n^2 + 97*n - 228)*x^2 - (16*n + 30)*x^3)*P[n - 1, x] + (x + (8*n + 45)*x^2 + (240*n + 451)*x^3 + (8*n + 15)*x^4)*D[P[n - 1, x], x])/(-n - (8*n^2 - 2*n)*x + (8*n^2 + 15*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n + (8*n^2 + 30*n - 30)*x + (120*n^2 + 97*n - 228)*x^2 - (16*n + 30)*x^3)*v[i - 1] + (x + (8*n + 45)*x^2 + (240*n + 451)*x^3 + (8*n + 15)*x^4)*deriv(v[i - 1], x))/(-n - (8*n^2 - 2*n)*x + (8*n^2 + 15*n)*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)

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-ee1a409b71674b97
• 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-ee1a409b71674b97
• 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-ee1a409b71674b97
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-ee1a409b71674b97

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 euler_polynomial, shift 0 e-ee1a409b71674b97

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Pascal-(1,7,1) array.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-a31f5c33917b15a7 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-7d90ede2da8763e3

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,7,1) array.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang_derivative_lag e-a31f5c33917b15a7 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.