A062145 → (θ+1)²

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:
   1: 1;
   2: 1,  16;
   3: 1,  40,    90;
   4: 1,  72,   405,    320;
   5: 1, 112,  1134,   2240,     875;
   6: 1, 160,  2520,   8960,    8750,    2016;
   7: 1, 216,  4860,  26880,   47250,   27216,    4116;
   8: 1, 280,  8505,  67200,  183750,  190512,   72030,    7680;
   9: 1, 352, 13860, 147840,  577500,  931392,  633864,  168960,  13365;
  10: 1, 432, 21384, 295680, 1559250, 3592512, 3803184, 1824768, 360855, 22000;

The row polynomials satisfy (n^2+n-2-(4*n^4+6*n^3-15*n^2-13*n+18)*x+(n^6+3*n^5-3*n^4-11*n^3+6*n^2+12*n-8)*x^2)*P(n,x) = (n^2+n-2-(4*n^4+6*n^3-27*n^2-17*n+10)*x+(n^6+3*n^5-24*n^4-55*n^3+22*n^2-39*n+14)*x^2+(3*n^6+15*n^5+14*n^4+3*n^3+36*n^2-23*n+6)*x^3)*P(n-1,x) + ((2*n+1)*x-(8*n^3+20*n^2-4*n+3)*x^2+(2*n^5+11*n^4+20*n^3+15*n^2+4*n-1)*x^3-(2*n^5+11*n^4+12*n^3-5*n^2+10*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n - 2 - (4*n^4 + 6*n^3 - 27*n^2 - 17*n + 10)*x + (n^6 + 3*n^5 - 24*n^4 - 55*n^3 + 22*n^2 - 39*n + 14)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*P[n - 1, x] + ((2*n + 1)*x - (8*n^3 + 20*n^2 - 4*n + 3)*x^2 + (2*n^5 + 11*n^4 + 20*n^3 + 15*n^2 + 4*n - 1)*x^3 - (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 - (4*n^4 + 6*n^3 - 15*n^2 - 13*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 + n - 2 - (4*n^4 + 6*n^3 - 27*n^2 - 17*n + 10)*x + (n^6 + 3*n^5 - 24*n^4 - 55*n^3 + 22*n^2 - 39*n + 14)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*v[i - 1] + ((2*n + 1)*x - (8*n^3 + 20*n^2 - 4*n + 3)*x^2 + (2*n^5 + 11*n^4 + 20*n^3 + 15*n^2 + 4*n - 1)*x^3 - (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 - (4*n^4 + 6*n^3 - 15*n^2 - 13*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• 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-42aebc790a856cde
• 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-42aebc790a856cde
• 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-42aebc790a856cde
• 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-42aebc790a856cde

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-42aebc790a856cde

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

  • premise (sequence): Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-62efc507a1cd208a root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

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.