A023531 → 2D+1+2n → Veronese 2 (j=1)

OEIS matches ?

• A154708: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A014601: 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: 0;
  1: 3;
  2: 4;
  3: 0, 7;
  4: 0, 8;
  5: 0, 0, 11;
  6: 0, 0, 12;
  7: 0, 0,  0, 15;
  8: 0, 0,  0, 16;
  9: 0, 0,  0,  0, 19;

The row polynomials satisfy P(n,x) = 2*x*P(n-2,x) - x^2*P(n-4,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 4, P(3,x) = 7*x.

P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 4;
P[3, x_] := 7*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 2, x] - x^2*P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 4; v[4] = 7*x; for(n = 4, N, my(i = n + 1); v[i] = 2*x*v[i - 2] - x^2*v[i - 4]); 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-611716aca3d048f2
• 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-611716aca3d048f2
• 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-611716aca3d048f2

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

• veronese_section r=2, j=1, shift 0 e-611716aca3d048f2

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

Same fingerprint (75)

Equal up to content, sign, reversal and a small index shift.

• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
• A010054 → reverse → D
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
• basis-map coefficients of A026729 → (θ+1)²
• basis-map coefficients of A021012 → D
• A135838 → γ
• A023531 → D → (θ+1)²
• A196347 → γ
• basis-map coefficients of A030528, shifted → reverse
• basis-map coefficients of A297191, shifted → (θ+1)²
• A130123 → reverse
• A136572 → θ+1
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• A130123 → (θ+1)²
• A010054 → reverse → θ+1
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
• A136572 → D
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
• basis-map coefficients of A030528, shifted → D
• A010054 → reverse → D
• A156992 → γ
• A085881 → γ
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
• A028326 → γ
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
• A023531 → 2D+1+2n → Veronese 2
• A356546 → γ
• A038338 → γ
• A130123 → D
• basis-map coefficients of A030528, shifted → (θ+1)²
• A344565 → γ
• A010054 → reverse → reverse
• A010054 → reverse → reverse
• A010054 → reverse → θ+1
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• basis-map coefficients of A021012 → (θ+1)²
• A010054 → reverse → θ+1
• basis-map coefficients of A021012 → θ+1
• A133080 → Veronese 2
• A178820 → γ
• A130123 → θ+1
• A023531 → (θ+1)² → reverse
• A131431: 3n + 1 preceded by n zeros.
• A038221 → γ
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• basis-map coefficients of A030528, shifted → θ+1
• A023531 → (θ+1)² → θ+1
• basis-map coefficients of A112227 → (θ+1)²
• basis-map coefficients of A026729, shifted → reverse
• basis-map coefficients of A021012 → reverse
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• A038325 → γ
• A007318 → D → γ
• A178821 → γ
• A023531 → (θ+1)² → (θ+1)²
• A178822 → γ
• basis-map coefficients of A112227 → θ+1
• A136572 → (θ+1)²
• A023531 → θ+1
• A010054 → reverse → D
• basis-map coefficients of A297191, shifted → D
• basis-map coefficients of A112227 → D
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A010054 → reverse → (θ+1)²
• A023531 → (θ+1)² → D
• basis-map coefficients of A026729, shifted → θ+1
• basis-map coefficients of A scaled Hermite triangle.
• A010054 → reverse → (θ+1)²
• basis-map coefficients of A026729, shifted → (θ+1)²
• basis-map coefficients of A026729 → D
• basis-map coefficients of A297191, shifted → θ+1
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• basis-map coefficients of A026729, shifted → D

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