A066325 → 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:
   1:    0,     1;
   2:   -1,    -1,     1;
   3:    2,    -2,    -2,    1;
   4:    0,     8,    -3,   -3,    1;
   5:   -8,    -2,    20,   -4,   -4,     1;
   6:   10,   -46,   -10,   40,   -5,    -5,    1;
   7:   36,    76,  -156,  -30,   70,    -6,   -6,   1;
   8: -112,   272,   322, -406,  -70,   112,   -7,  -7,  1;
   9: -160, -1028,  1168, 1008, -896,  -140,  168,  -8, -8,  1;
  10: 1188, -1468, -5220, 3744, 2604, -1764, -252, 240, -9, -9, 1;

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

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, 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] = x; for(n = 2, N, my(i = n); v[i] = -(1 - x)*v[i - 1] - 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-7d57ae2f42078672

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-7d57ae2f42078672

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

  • premise (sequence): Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift -1 e-9c6e37fd70f39446

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): hermite_he
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
        • theorem_check three_term_recurrence e-5cce3c36fe7dc594 root: theorem check

          Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

        • preserver_image euler_polynomial, shift 0 e-74d857940dd4e27e

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

          • premise (sequence): hermite_he
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
            • ↵ theorem_check three_term_recurrence e-5cce3c36fe7dc594 (expanded above)
            • basis_map A066325 e-b31a65521e9db5ca

              Hermite-type basis map [MW08, Thm 2] polytool ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d

                  Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?

                  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

                      Parent-row provers of polytool polytool ?

                    • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
                • parent_provers periodic_product e-b7eca039219a7f53 root: theorem check

                  Parent-row provers of polytool polytool ?

                • preserver_image mobius, shift 0 e-7e44b66e04c497d6

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                    • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
        • ↵ basis_map A066325 e-b31a65521e9db5ca (expanded above)
    • basis_map A159834 e-c585175918782f17

      Hermite-type basis map [MW08, Thm 2] 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 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)
  • premise (sequence): Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    • ↵ preserver_image affine, shift -1 e-9c6e37fd70f39446 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
4. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
5. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
6. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
7. [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
8. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.
9. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
10. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023