A023531 → 2D+1+2n → θ+1

OEIS matches ?

• A033951: 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, 6;
  2: 0, 8, 15;
  3: 0, 0, 18, 28;
  4: 0, 0,  0, 32, 45;
  5: 0, 0,  0,  0, 50, 66;
  6: 0, 0,  0,  0,  0, 72, 91;
  7: 0, 0,  0,  0,  0,  0, 98, 120;
  8: 0, 0,  0,  0,  0,  0,  0, 128, 153;
  9: 0, 0,  0,  0,  0,  0,  0,   0, 162, 190;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2*x + (2*n^2 + 3*n + 1)*x^2)*P[n - 1, x])/(2*n^2 - 4*n + 2 + (2*n^2 - 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] = ((2*n^2*x + (2*n^2 + 3*n + 1)*x^2)*v[i - 1])/(2*n^2 - 4*n + 2 + (2*n^2 - 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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-9cac8cec827e3b4d
• 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-9cac8cec827e3b4d
• 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-9cac8cec827e3b4d

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-9cac8cec827e3b4d

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

  • premise (sequence): A natural number operator.
    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-3fa49e62bb44b5e9

      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)
    • preserver_image mobius, shift 0 e-cbd96c681ac3cbdc

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, (n-1) zeros followed by (2n, 1).
        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-224384a428b81acf

          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 (expanded above)
            • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (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.