A100861 → reverse → (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: 0, 4;
  2: 0, 4,  9;
  3: 0, 0, 27,  16;
  4: 0, 0, 27,  96,   25;
  5: 0, 0,  0, 240,  250,    36;
  6: 0, 0,  0, 240, 1125,   540,    49;
  7: 0, 0,  0,   0, 2625,  3780,  1029,    64;
  8: 0, 0,  0,   0, 2625, 15120, 10290,  1792,   81;
  9: 0, 0,  0,   0,    0, 34020, 61740, 24192, 2916, 100;

The row polynomials satisfy (n^4+6*n^3+13*n^2+12*n+4-(4*n^3+18*n^2+14*n-4)*x+4*n^2*x^2)*P(n,x) = -(n^5+5*n^4+7*n^3-n^2-8*n-4-(5*n^4+28*n^3+49*n^2+42*n+20)*x+(8*n^3+34*n^2+58*n+28)*x^2-(4*n^2+8*n+4)*x^3)*P(n-1,x) + ((2*n^4+12*n^3+26*n^2+24*n+8)*x-(8*n^3+44*n^2+68*n+32)*x^2+(8*n^2+16*n+8)*x^3)*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^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (5*n^4 + 28*n^3 + 49*n^2 + 42*n + 20)*x + (8*n^3 + 34*n^2 + 58*n + 28)*x^2 - (4*n^2 + 8*n + 4)*x^3)*P[n - 1, x] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (8*n^3 + 44*n^2 + 68*n + 32)*x^2 + (8*n^2 + 16*n + 8)*x^3)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (4*n^3 + 18*n^2 + 14*n - 4)*x + 4*n^2*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^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (5*n^4 + 28*n^3 + 49*n^2 + 42*n + 20)*x + (8*n^3 + 34*n^2 + 58*n + 28)*x^2 - (4*n^2 + 8*n + 4)*x^3)*v[i - 1] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (8*n^3 + 44*n^2 + 68*n + 32)*x^2 + (8*n^2 + 16*n + 8)*x^3)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (4*n^3 + 18*n^2 + 14*n - 4)*x + 4*n^2*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-38acf07bf4ec9f74
• 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-38acf07bf4ec9f74
• 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-38acf07bf4ec9f74

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-38acf07bf4ec9f74

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

  • premise (sequence): Exponential Riordan array (1, x(1+x/2)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-01d50252f4aaa594

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • gamma_polynomial scale 1, shift 0 e-037c1a08736a9fd9

          Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
            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-e7942008ad3e1420 root: theorem check

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

            • preserver_image reversal, shift 0 e-7078061f6d32a158

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
                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-e7942008ad3e1420 (expanded above)
        • parent_provers paper_theorem e-8187ad69de0e901a root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-a3a653eabafbad6b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Exponential Riordan array (1, x(1+x/2)).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            no recorded property covers this requirement

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.