A007318 → ×(−(1/2)n+(1/2)n²) → θ+1

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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:  0;
  2:  1,   4,    3;
  3:  3,  18,   27,    12;
  4:  6,  48,  108,    96,    30;
  5: 10, 100,  300,   400,   250,    60;
  6: 15, 180,  675,  1200,  1125,   540,   105;
  7: 21, 294, 1323,  2940,  3675,  2646,  1029,   168;
  8: 28, 448, 2352,  6272,  9800,  9408,  5488,  1792,  252;
  9: 36, 648, 3888, 12096, 22680, 27216, 21168, 10368, 2916, 360;

The row polynomials satisfy (1+(3*n-3)*x+(3*n^2-6*n+2)*x^2+(n^3-3*n^2+2*n)*x^3)*P(n,x) = (1+(3*n+3)*x+(3*n^2+9*n+2)*x^2+(n^3+9*n^2+8*n-2)*x^3+(3*n^3+9*n^2-3)*x^4+(3*n^3+3*n^2-3*n-1)*x^5+(n^3-n)*x^6)*P(n-3,x) - (3+(9*n+3)*x+(9*n^2+12*n-9)*x^2+(3*n^3+15*n^2-9*n-15)*x^3+(6*n^3+3*n^2-18*n-6)*x^4+(3*n^3-3*n^2-6*n)*x^5)*P(n-2,x) + (3+(9*n-3)*x+(9*n^2-3*n-9)*x^2+(3*n^3+3*n^2-15*n+3)*x^3+(3*n^3-6*n^2-3*n+6)*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1 + 4*x + 3*x^2.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1 + 4*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n + 3)*x + (3*n^2 + 9*n + 2)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^3 + 9*n^2 - 3)*x^4 + (3*n^3 + 3*n^2 - 3*n - 1)*x^5 + (n^3 - n)*x^6)*P[n - 3, x] - (3 + (9*n + 3)*x + (9*n^2 + 12*n - 9)*x^2 + (3*n^3 + 15*n^2 - 9*n - 15)*x^3 + (6*n^3 + 3*n^2 - 18*n - 6)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*P[n - 2, x] + (3 + (9*n - 3)*x + (9*n^2 - 3*n - 9)*x^2 + (3*n^3 + 3*n^2 - 15*n + 3)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*P[n - 1, x])/(1 + (3*n - 3)*x + (3*n^2 - 6*n + 2)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1 + 4*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((1 + (3*n + 3)*x + (3*n^2 + 9*n + 2)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^3 + 9*n^2 - 3)*x^4 + (3*n^3 + 3*n^2 - 3*n - 1)*x^5 + (n^3 - n)*x^6)*v[i - 3] - (3 + (9*n + 3)*x + (9*n^2 + 12*n - 9)*x^2 + (3*n^3 + 15*n^2 - 9*n - 15)*x^3 + (6*n^3 + 3*n^2 - 18*n - 6)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*v[i - 2] + (3 + (9*n - 3)*x + (9*n^2 - 3*n - 9)*x^2 + (3*n^3 + 3*n^2 - 15*n + 3)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*v[i - 1])/(1 + (3*n - 3)*x + (3*n^2 - 6*n + 2)*x^2 + (n^3 - 3*n^2 + 2*n)*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-21ef48566d5646a5
• 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-21ef48566d5646a5
• 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-21ef48566d5646a5

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-21ef48566d5646a5

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

  • premise (sequence): Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-a72beb3d57d8ea71

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
        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-ccfad80d3bf960c1

          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)
    • ↵ preserver_image euler_polynomial, shift 0 e-ccfad80d3bf960c1 (expanded above)

Related sequences ?

Same fingerprint (27)

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

• A178821 → θ+1
• A135838 → D+1
• A178822 → D+1
• A085880 → D+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A007318 → ×(1+(3/2)n+(1/2)n²) → D+1
• A085880 → θ+1
• A028326 → D+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → θ+1
• A178820 → θ+1
• A028326 → θ+1
• A135838 → θ+1
• A007318 → D → θ+1
• basis-map coefficients of A113214, shifted → D+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A038208 → D+1
• A178820 → D+1
• A344565 → θ+1
• A007318 → ×(1+2n+n²) → D+1
• A178821 → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A178822 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A038208 → θ+1
• A007318 → D → D+1
• A007318 → ×(1+2n+n²) → θ+1
• A344565 → D+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. [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.