A133087 → (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:
  0: 1;
  1: 2,  4;
  2: 1,  8,   9;
  3: 2, 20,  36,   16;
  4: 1, 16,  54,   64,   25;
  5: 2, 36, 144,  224,  150,   36;
  6: 1, 24, 135,  320,  375,  216,   49;
  7: 2, 52, 324,  880, 1250,  972,  392,   64;
  8: 1, 32, 252,  896, 1750, 2016, 1372,  512,  81;
  9: 2, 68, 576, 2240, 4900, 6552, 5488, 2816, 810, 100;

The row polynomials satisfy (4+(10*n-4)*x+(14*n^2-18*n+4)*x^2+(6*n^3-8*n^2)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = -(12*n-20+(51*n^2-93*n+14)*x+(60*n^3-110*n^2+22*n+4)*x^2+(24*n^4-18*n^3-48*n^2+18*n+28)*x^3+(3*n^5+9*n^4-25*n^3-3*n^2+16*n+8)*x^4+(2*n^5-3*n^4-2*n^3+4*n^2+n-2)*x^5)*P(n-1,x) + (3+(18*n-4)*x+(33*n^2+n-11)*x^2+(18*n^3+40*n^2-30*n-12)*x^3+(3*n^4+28*n^3+2*n^2-19*n-9)*x^4+(5*n^4+10*n^3-7*n^2-6*n)*x^5+(2*n^4-2*n^2+1)*x^6)*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[(-(12*n - 20 + (51*n^2 - 93*n + 14)*x + (60*n^3 - 110*n^2 + 22*n + 4)*x^2 + (24*n^4 - 18*n^3 - 48*n^2 + 18*n + 28)*x^3 + (3*n^5 + 9*n^4 - 25*n^3 - 3*n^2 + 16*n + 8)*x^4 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x^5)*P[n - 1, x] + (3 + (18*n - 4)*x + (33*n^2 + n - 11)*x^2 + (18*n^3 + 40*n^2 - 30*n - 12)*x^3 + (3*n^4 + 28*n^3 + 2*n^2 - 19*n - 9)*x^4 + (5*n^4 + 10*n^3 - 7*n^2 - 6*n)*x^5 + (2*n^4 - 2*n^2 + 1)*x^6)*D[P[n - 1, x], x])/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)]];
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] = (-(12*n - 20 + (51*n^2 - 93*n + 14)*x + (60*n^3 - 110*n^2 + 22*n + 4)*x^2 + (24*n^4 - 18*n^3 - 48*n^2 + 18*n + 28)*x^3 + (3*n^5 + 9*n^4 - 25*n^3 - 3*n^2 + 16*n + 8)*x^4 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x^5)*v[i - 1] + (3 + (18*n - 4)*x + (33*n^2 + n - 11)*x^2 + (18*n^3 + 40*n^2 - 30*n - 12)*x^3 + (3*n^4 + 28*n^3 + 2*n^2 - 19*n - 9)*x^4 + (5*n^4 + 10*n^3 - 7*n^2 - 6*n)*x^5 + (2*n^4 - 2*n^2 + 1)*x^6)*deriv(v[i - 1], x))/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)); 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-7c04308011ed96cd
• 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-7c04308011ed96cd
• 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-7c04308011ed96cd

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-7c04308011ed96cd

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

  • premise (sequence): A133080 * A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A007318 e-3537980ff71b1a89

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-aa3a6b002de14e0c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-28a1cf3333f63fa4

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-4d02ac33f5b2fbc9 (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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.