A008278 → x ↦ x+1 (shift +1) → Pₙ + Pₙ₊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:      3,       1;
  1:      7,       6,        1;
  2:     20,      28,       11,        1;
  3:     67,     132,       86,       20,        1;
  4:    255,     653,      607,      243,       37,       1;
  5:   1080,    3420,     4172,     2428,      667,      70,       1;
  6:   5017,   18989,    28849,    22293,     9102,    1818,     135,      1;
  7:  25287,  111608,   203772,   197720,   108642,   33080,    4972,    264,     1;
  8: 137122,  692650,  1481015,  1737419,  1209644,  503158,  118645,  13717,   521,    1;
  9: 794545, 4526398, 11114639, 15340444, 12997273, 6917650, 2264957, 424160, 38273, 1034, 1;

The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-2,x) - (1+2*x+x^2)*d/dx P(n-2,x) + (n+1+(n+1)*x)*P(n-1,x) - (1+2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + x, P(1,x) = 7 + 6*x + x^2.

P[0, x_] := 3 + x;
P[1, x_] := 7 + 6*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + (n - 1)*x)*P[n - 2, x] - (1 + 2*x + x^2)*D[P[n - 2, x], x] + (n + 1 + (n + 1)*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; v[2] = 7 + 6*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (n + (n - 1)*x)*v[i - 2] - (1 + 2*x + x^2)*deriv(v[i - 2], x) + (n + 1 + (n + 1)*x)*v[i - 1] - (1 + 2*x + x^2)*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 - 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-7a81e5633c7cc19a

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-7a81e5633c7cc19a

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

  • premise (sequence): Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 1 e-236d0fb15dc6eca0

      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 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 (expanded above)
  • premise (sequence): Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 1 e-236d0fb15dc6eca0 (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.