A007318 → 2θ+1 → (θ+1)²

OEIS matches ?

No OEIS matches.

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: 1,  12;
  2: 1,  24,   45;
  3: 1,  36,  135,  112;
  4: 1,  48,  270,  448,   225;
  5: 1,  60,  450, 1120,  1125,   396;
  6: 1,  72,  675, 2240,  3375,  2376,   637;
  7: 1,  84,  945, 3920,  7875,  8316,  4459,   960;
  8: 1,  96, 1260, 6272, 15750, 22176, 17836,  7680,  1377;
  9: 1, 108, 1620, 9408, 28350, 49896, 53508, 34560, 12393, 1900;

The row polynomials satisfy (1+(11*n-8)*x+(11*n^2-11*n+3)*x^2+(2*n^3-n^2)*x^3)*P(n,x) = (1+(11*n+4)*x+(11*n^2+22*n+6)*x^2+(2*n^3+16*n^2+15*n+4)*x^3+(2*n^3+5*n^2+4*n+1)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (11*n + 4)*x + (11*n^2 + 22*n + 6)*x^2 + (2*n^3 + 16*n^2 + 15*n + 4)*x^3 + (2*n^3 + 5*n^2 + 4*n + 1)*x^4)*P[n - 1, x])/(1 + (11*n - 8)*x + (11*n^2 - 11*n + 3)*x^2 + (2*n^3 - n^2)*x^3)]];
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] = ((1 + (11*n + 4)*x + (11*n^2 + 22*n + 6)*x^2 + (2*n^3 + 16*n^2 + 15*n + 4)*x^3 + (2*n^3 + 5*n^2 + 4*n + 1)*x^4)*v[i - 1])/(1 + (11*n - 8)*x + (11*n^2 - 11*n + 3)*x^2 + (2*n^3 - n^2)*x^3)); 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-e317db6457cebe23
• 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-e317db6457cebe23
• 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-e317db6457cebe23

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-e317db6457cebe23

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

  • premise (sequence): Triangle, row sums = number of edges in n-dimensional hypercubes.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 0 e-c4ea45619bc7f60a

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

                  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)

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