A008277 → θ+2 (shift +1) → D

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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:    0;
  1:    3;
  2:    9,     8;
  3:   21,    48,     15;
  4:   45,   200,    150,      24;
  5:   93,   720,    975,     360,     35;
  6:  189,  2408,   5250,    3360,    735,     48;
  7:  381,  7728,  25515,   25200,   9310,   1344,    63;
  8:  765, 24200, 116550,  166824,  92610,  22176,  2268,   80;
  9: 1533, 74640, 511575, 1020600, 798945, 282240, 47250, 3600, 99;

The row polynomials satisfy 4*P(n,x) = -(4+10*x+18*x^2-4*x^3)*P(n-3,x) + (9+x-17*x^2)*d/dx P(n-3,x) + (3*x+x^2-4*x^3)*d^2/dx^2 P(n-3,x) + (18+12*x-12*x^2)*P(n-2,x) + (6+16*x)*d/dx P(n-2,x) + (3*x+4*x^2)*d^2/dx^2 P(n-2,x) + (6+12*x)*P(n-1,x) - 3*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 9 + 8*x.

P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 9 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4 + 10*x + 18*x^2 - 4*x^3)*P[n - 3, x] + (9 + x - 17*x^2)*D[P[n - 3, x], x] + (3*x + x^2 - 4*x^3)*D[P[n - 3, x], {x, 2}] + (18 + 12*x - 12*x^2)*P[n - 2, x] + (6 + 16*x)*D[P[n - 2, x], x] + (3*x + 4*x^2)*D[P[n - 2, x], {x, 2}] + (6 + 12*x)*P[n - 1, x] - 3*D[P[n - 1, x], x])/(4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 9 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(4 + 10*x + 18*x^2 - 4*x^3)*v[i - 3] + (9 + x - 17*x^2)*deriv(v[i - 3], x) + (3*x + x^2 - 4*x^3)*deriv(deriv(v[i - 3], x), x) + (18 + 12*x - 12*x^2)*v[i - 2] + (6 + 16*x)*deriv(v[i - 2], x) + (3*x + 4*x^2)*deriv(deriv(v[i - 2], x), x) + (6 + 12*x)*v[i - 1] - 3*deriv(v[i - 1], 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-967e3499abfd84e8
• 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-967e3499abfd84e8
• 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-967e3499abfd84e8

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 derivative_polynomial, shift 0 e-967e3499abfd84e8

  Laguerre’s theorem on differential operators [SP14] 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.