A106800 → reverse (shift +1) → (θ+1)²

OEIS matches ?

• A078468: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 0, 4;
  2: 0, 4,    9;
  3: 0, 4,   27,    16;
  4: 0, 4,   63,    96,     25;
  5: 0, 4,  135,   400,    250,     36;
  6: 0, 4,  279,  1440,   1625,    540,     49;
  7: 0, 4,  567,  4816,   8750,   5040,   1029,    64;
  8: 0, 4, 1143, 15456,  42525,  37800,  13034,  1792,   81;
  9: 0, 4, 2295, 48400, 194250, 250236, 129654, 29568, 2916, 100;

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

P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 2, x] + 4*x^3*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] - (1 - 4*x)*P[n - 1, x] + (2*x - 4*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] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = 9*x^2*v[i - 2] + 4*x^3*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) - (1 - 4*x)*v[i - 1] + (2*x - 4*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-aa3ce102f535b7a2
• 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-aa3ce102f535b7a2
• 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-aa3ce102f535b7a2

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 euler_polynomial, shift 0 e-aa3ce102f535b7a2

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

  • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A008277 e-857a86cbb6c7881e

      Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

      • premise (sequence): basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
        • preserver_image affine, shift 0 e-0ba23af98afc4615

          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

              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)
        • preserver_image affine, shift 0 e-8c1f314a0c6d10ae

          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)
            • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    • preserver_image reversal, shift 1 e-ad13a5d17ca116d8

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= 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-2ab1dfe057e84c9d root: theorem check

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

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. [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
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.