A038221 → Pₙ + Pₙ₊₁

OEIS matches ?

• A167898: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168732: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168684: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A003949: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168780: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168876: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168828: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168924: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:     4,      3;
  1:    12,     21,       9;
  2:    36,     99,      90,      27;
  3:   108,    405,     567,     351,       81;
  4:   324,   1539,    2916,    2754,     1296,      243;
  5:   972,   5589,   13365,   17010,    12150,     4617,      729;
  6:  2916,  19683,   56862,   91125,    87480,    50301,    16038,    2187;
  7:  8748,  67797,  229635,  443961,   535815,   413343,   199017,   54675,    6561;
  8: 26244, 229635,  892296, 2020788,  2939328,  2847474,  1837080,  761076,  183708,  19683;
  9: 78732, 767637, 3365793, 8739252, 14880348, 17360406, 14053662, 7794468, 2834352, 610173, 59049;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (3 + 3*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-cf7bdd46272d1037

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-cf7bdd46272d1037

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-b7eead465111f7d4 root: theorem check

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

    • parent_provers periodic_product e-110e02c529078258 root: theorem check

      Parent-row provers of polytool polytool ?

    • basis_map A013610 e-d377a44ea8f9b38f

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-0d085d05819c04f2 root: theorem check

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

        • preserver_image reversal, shift 0 e-045f9d56b54ceaf2

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of coefficients in expansion of (1+2*x)^n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-a6866143cc223753 root: theorem check

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

            • basis_map A013609 e-41b2234a78052e4f

              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)
            • gamma_polynomial scale 1, shift 0 e-4c4f74d9ba733cff

              Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

              • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-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-eab26daf7024ea74 root: theorem check

                  Parent-row provers of polytool polytool ?

            • preserver_image reversal, shift 0 e-6d15baf5e54583ee

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-0d085d05819c04f2 (expanded above)
                • parent_provers periodic_product e-1697ba43e31b7729 root: theorem check

                  Parent-row provers of polytool polytool ?

                • basis_map A007318 e-4608a916413bc814

                  Geometric basis map (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 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 (expanded above)
                            • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                            • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (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)
                • preserver_image affine, shift 0 e-17c52917511dc051

                  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)
                    • ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (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)
            • parent_provers periodic_product e-d08b7f9df01d4f3c root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image mobius, shift 0 e-2c323256b27180b8

              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)
                • ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (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)
        • ↵ parent_provers periodic_product e-1697ba43e31b7729 (expanded above)
        • ↵ basis_map A007318 e-4608a916413bc814 (expanded above)
        • ↵ preserver_image affine, shift 0 e-17c52917511dc051 (expanded above)
  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

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. [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.