A081580 → 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, 14,    3;
  3: 1, 26,   39,     4;
  4: 1, 38,  183,    76,     5;
  5: 1, 50,  435,   580,   125,      6;
  6: 1, 62,  795,  2380,  1325,    186,     7;
  7: 1, 74, 1263,  6340,  7925,   2526,   259,    8;
  8: 1, 86, 1839, 13324, 30725,  19986,  4291,  344,   9;
  9: 1, 98, 2523, 24196, 87005, 104406, 42343, 6728, 441, 10;

The row polynomials satisfy -(n+(6*n^2-2*n)*x-(6*n^2+11*n)*x^2)*P(n,x) = -(n+(6*n^2+22*n-22)*x+(66*n^2+49*n-124)*x^2-(12*n+22)*x^3)*P(n-1,x) + (x+(6*n+33)*x^2+(132*n+243)*x^3+(6*n+11)*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 + (6*n^2 + 22*n - 22)*x + (66*n^2 + 49*n - 124)*x^2 - (12*n + 22)*x^3)*P[n - 1, x] + (x + (6*n + 33)*x^2 + (132*n + 243)*x^3 + (6*n + 11)*x^4)*D[P[n - 1, x], x])/(-n - (6*n^2 - 2*n)*x + (6*n^2 + 11*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 + (6*n^2 + 22*n - 22)*x + (66*n^2 + 49*n - 124)*x^2 - (12*n + 22)*x^3)*v[i - 1] + (x + (6*n + 33)*x^2 + (132*n + 243)*x^3 + (6*n + 11)*x^4)*deriv(v[i - 1], x))/(-n - (6*n^2 - 2*n)*x + (6*n^2 + 11*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-5726a07dd01f8b02
• 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-5726a07dd01f8b02
• 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-5726a07dd01f8b02
• 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-5726a07dd01f8b02

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-5726a07dd01f8b02

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

  • premise (sequence): Pascal-(1,5,1) 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-307703197d4a988b root: theorem check

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

    • preserver_image reversal, shift 0 e-bf7624d7e60440c3

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,5,1) 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-307703197d4a988b (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.