A143494 → Pₙ + Pₙ₊₁

OEIS matches ?

• A078468: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:    3,     1;
   2:    6,     6,      1;
   3:   12,    24,     10,       1;
   4:   24,    84,     64,      15,       1;
   5:   48,   276,    340,     139,      21,      1;
   6:   96,   876,   1636,    1035,     265,     28,      1;
   7:  192,  2724,   7420,    6811,    2625,    461,     36,     1;
   8:  384,  8364,  32404,   41475,   22561,   5852,    749,    45,    1;
   9:  768, 25476, 137980,  239779,  176841,  63525,  11844,  1154,   55,  1;
  10: 1536, 77196, 577396, 1336875, 1300825, 621516, 158277, 22230, 1704, 66, 1;

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

P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 3 + x; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f5ad2d4d1fdd1ba8

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

• obreschkoff_combination e-f5ad2d4d1fdd1ba8

  Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

  • premise (sequence): Triangle read by rows: 2-Stirling numbers of the second kind.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-d198e25506383784 root: theorem check

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

    • preserver_image derivative_polynomial, shift 0 e-34c9c7195bdd20aa

      Laguerre’s theorem on differential operators [SP14] 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)
  • premise (sequence): Triangle read by rows: 2-Stirling numbers of the second kind.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    • ↵ preserver_image derivative_polynomial, shift 0 e-34c9c7195bdd20aa (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
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. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.