A023531 → (1/2)D²+2D+1 → θ+1

OEIS matches ?

• A006000: 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: 2, 2;
  2: 1, 8,  3;
  3: 0, 6, 18,  4;
  4: 0, 0, 18, 32,  5;
  5: 0, 0,  0, 40, 50,   6;
  6: 0, 0,  0,  0, 75,  72,   7;
  7: 0, 0,  0,  0,  0, 126,  98,   8;
  8: 0, 0,  0,  0,  0,   0, 196, 128,   9;
  9: 0, 0,  0,  0,  0,   0,   0, 288, 162, 10;

The row polynomials satisfy P(n,x) = -x^4*P(n-4,x) + 4*x^3*P(n-3,x) - 6*x^2*P(n-2,x) + 4*x*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 1 + 8*x + 3*x^2, P(3,x) = 6*x + 18*x^2 + 4*x^3.

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 1 + 8*x + 3*x^2;
P[3, x_] := 6*x + 18*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^4*P[n - 4, x] + 4*x^3*P[n - 3, x] - 6*x^2*P[n - 2, x] + 4*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 1 + 8*x + 3*x^2; v[4] = 6*x + 18*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -x^4*v[i - 4] + 4*x^3*v[i - 3] - 6*x^2*v[i - 2] + 4*x*v[i - 1]); 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-ac5d36b14b8fced4
• 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-ac5d36b14b8fced4
• 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-ac5d36b14b8fced4

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-ac5d36b14b8fced4

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

  • premise (sequence): Number triangle T(n,k)=C(n,k)C(2,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image derivative_polynomial, shift 0 e-0c9044372fd705ed

      Laguerre’s theorem on differential operators [SP14] 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)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.