A008277 → θ+2 (shift +1) → 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: 2;
  1: 2,    6;
  2: 2,   18,     12;
  3: 2,   42,     72,     20;
  4: 2,   90,    300,    200,      30;
  5: 2,  186,   1080,   1300,     450,     42;
  6: 2,  378,   3612,   7000,    4200,    882,     56;
  7: 2,  762,  11592,  34020,   31500,  11172,   1568,    72;
  8: 2, 1530,  36300, 155400,  208530, 111132,  25872,  2592,   90;
  9: 2, 3066, 111960, 682100, 1275750, 958734, 329280, 54000, 4050, 110;

The row polynomials satisfy P(n,x) = (3*x-6*x^2+x^3)*P(n-3,x) + (2*x-3*x^2)*d/dx P(n-3,x) + (x^2-x^3)*d^2/dx^2 P(n-3,x) + (1+6*x-3*x^2)*P(n-2,x) + 2*x*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) + 3*x*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 2 + 6*x, P(2,x) = 2 + 18*x + 12*x^2.

P[0, x_] := 2;
P[1, x_] := 2 + 6*x;
P[2, x_] := 2 + 18*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(3*x - 6*x^2 + x^3)*P[n - 3, x] + (2*x - 3*x^2)*D[P[n - 3, x], x] + (x^2 - x^3)*D[P[n - 3, x], {x, 2}] + (1 + 6*x - 3*x^2)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}] + 3*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + 6*x; v[3] = 2 + 18*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3*x - 6*x^2 + x^3)*v[i - 3] + (2*x - 3*x^2)*deriv(v[i - 3], x) + (x^2 - x^3)*deriv(deriv(v[i - 3], x), x) + (1 + 6*x - 3*x^2)*v[i - 2] + 2*x*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), x) + 3*x*v[i - 1]); 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-d901b9327c026219
• 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-d901b9327c026219
• 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-d901b9327c026219

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-d901b9327c026219

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

  • premise (sequence): Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 1 e-4ce9900b2631a004

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

      • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-995393ce4e49a8d6 root: theorem check

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

        • basis_map A008277 e-c52960bc5e97caa0

          Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image mobius, shift 0 e-ad67990681a86223

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)
        • preserver_image reversal, shift 0 e-d527982dc7c0aa3b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-8e580efc86a38a12 root: theorem check

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

            • preserver_image reversal, shift 0 e-3256729839b26765

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-995393ce4e49a8d6 (expanded above)
                • ↵ basis_map A008277 e-c52960bc5e97caa0 (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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.