A266213 → 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: 1,  4;
  3: 1,  8,   6;
  4: 1, 12,  24,    8;
  5: 1, 16,  54,   48,   10;
  6: 1, 20,  96,  152,   80,   12;
  7: 1, 24, 150,  352,  330,  120,   14;
  8: 1, 28, 216,  680,  960,  612,  168,  16;
  9: 1, 32, 294, 1168, 2250, 2160, 1022, 224, 18;

The row polynomials satisfy -(n^2+n+(2*n^3-n^2+n+4)*x-(2*n^3-n^2-3*n-6)*x^2-(n^2-3*n+2)*x^3)*P(n,x) = -(n^2+n+(2*n^3+7*n^2+n-4)*x+(6*n^3+7*n^2+n-6)*x^2+(n^2-3*n+2)*x^3)*P(n-1,x) + ((n+1)*x+(2*n^2+10*n+9)*x^2+(12*n^2+26*n+19)*x^3+(2*n^2+10*n+3)*x^4+n*x^5)*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^2 + n + (2*n^3 + 7*n^2 + n - 4)*x + (6*n^3 + 7*n^2 + n - 6)*x^2 + (n^2 - 3*n + 2)*x^3)*P[n - 1, x] + ((n + 1)*x + (2*n^2 + 10*n + 9)*x^2 + (12*n^2 + 26*n + 19)*x^3 + (2*n^2 + 10*n + 3)*x^4 + n*x^5)*D[P[n - 1, x], x])/(-n^2 - n - (2*n^3 - n^2 + n + 4)*x + (2*n^3 - n^2 - 3*n - 6)*x^2 + (n^2 - 3*n + 2)*x^3)]];
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^2 + n + (2*n^3 + 7*n^2 + n - 4)*x + (6*n^3 + 7*n^2 + n - 6)*x^2 + (n^2 - 3*n + 2)*x^3)*v[i - 1] + ((n + 1)*x + (2*n^2 + 10*n + 9)*x^2 + (12*n^2 + 26*n + 19)*x^3 + (2*n^2 + 10*n + 3)*x^4 + n*x^5)*deriv(v[i - 1], x))/(-n^2 - n - (2*n^3 - n^2 + n + 4)*x + (2*n^3 - n^2 - 3*n - 6)*x^2 + (n^2 - 3*n + 2)*x^3)); 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-1b28fe7d21cc4e12
• 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-1b28fe7d21cc4e12
• 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-1b28fe7d21cc4e12
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-1b28fe7d21cc4e12, e-46d00cd0dd655b47
• Real-rooted. Every row P_n, n >= 1, 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-46d00cd0dd655b47
• Coefficients of one sign. Every nonzero row, n >= 1: 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-46d00cd0dd655b47
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-46d00cd0dd655b47

Proof chains

2 incoming proof steps. 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-1b28fe7d21cc4e12

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

  • premise (sequence): Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…
    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-1e993b9b3c7faf34 root: theorem check

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

• preserver_image euler_polynomial, shift 0 e-46d00cd0dd655b47

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

  • premise (sequence): Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-f5199491b1877911 root: theorem check

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

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.