A007318 → Pₙ + Pₙ₊₁

OEIS matches ?

• A122391: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168728: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167881: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A003945: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A098011: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A042950: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168680: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A007283: 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: 2,  1;
  1: 2,  3,  1;
  2: 2,  5,  4,   1;
  3: 2,  7,  9,   5,   1;
  4: 2,  9, 16,  14,   6,   1;
  5: 2, 11, 25,  30,  20,   7,   1;
  6: 2, 13, 36,  55,  50,  27,   8,   1;
  7: 2, 15, 49,  91, 105,  77,  35,   9,  1;
  8: 2, 17, 64, 140, 196, 182, 112,  44, 10,  1;
  9: 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-c4dc09241b591172

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

• obreschkoff_combination e-c4dc09241b591172

  Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

  • premise (sequence): binomial
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-317eb4ccca60c6ac root: theorem check

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

    • preserver_image reversal, shift 0 e-033cf842aad3bdf7

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): binomial
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
        • basis_map A007318 e-095baa2a9dfe188f

          Geometric basis map (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)
        • parent_provers periodic_product e-13983862a411779c root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-989c95e5c893fce9

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1-x,x/(1-x)).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers periodic_product e-b62074ba45057e82 root: theorem check

              Parent-row provers of polytool polytool ?

        • gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea

          Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

          • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • preserver_image reversal, shift 0 e-0e058c428dfdfeb6

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check

                  Parent-row provers of polytool polytool ?

            • ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)
        • preserver_image affine, shift 0 e-3a52de7742ed9314

          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)
    • ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
    • ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
    • ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
    • ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
    • ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
  • premise (sequence): binomial
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

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

• A028326 → Pₙ + Pₙ₊₁

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. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
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. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.