A066325 → D+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:    1;
  1:    1,    1;
  2:   -1,    2,     1;
  3:   -3,   -3,     3,     1;
  4:    3,  -12,    -6,     4,    1;
  5:   15,   15,   -30,   -10,    5,    1;
  6:  -15,   90,    45,   -60,  -15,    6,    1;
  7: -105, -105,   315,   105, -105,  -21,    7,   1;
  8:  105, -840,  -420,   840,  210, -168,  -28,   8, 1;
  9:  945,  945, -3780, -1260, 1890,  378, -252, -36, 9, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 1)*x + x^2)*v[i - 1] - (n + 1 + x)*deriv(v[i - 1], x))/(n + 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-f8533e45cc68d6b0
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-f8533e45cc68d6b0

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-f8533e45cc68d6b0

  Laguerre’s theorem on differential operators [SP14] 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)

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. [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
7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
8. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023