A099097 → Veronese 2

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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:
   1: 1;
   2: 0;
   3: 0, 9;
   4: 0, 6;
   5: 0, 1,  81;
   6: 0, 0, 108;
   7: 0, 0,  54,  729;
   8: 0, 0,  12, 1458;
   9: 0, 0,   1, 1215,  6561;
  10: 0, 0,   0,  540, 17496;

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

P[1, x_] := 1;
P[2, x_] := 0;
P[3, x_] := 9*x;
P[4, x_] := 6*x;
P[n_, x_] := P[n, x] = Expand[9*x*P[n - 2, x] + 6*x*P[n - 3, x] + x*P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; v[3] = 9*x; v[4] = 6*x; for(n = 5, N, my(i = n); v[i] = 9*x*v[i - 2] + 6*x*v[i - 3] + x*v[i - 4]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-999bd295896592e8
• Coefficients of one sign. Every nonzero row, n >= 1: 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-999bd295896592e8
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-999bd295896592e8

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=0, shift 0 e-999bd295896592e8

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

  • premise (sequence): Riordan array (1, 3+x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A099097 e-5ba8fe4021f85333

      Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
        no recorded property covers this requirement
    • theorem_check ma_wang_interval e-09c939a271fe525a root: theorem check

      Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

    • basis_map A038291 e-2d0268d4f6f3cbee

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 9]
        • preserver_image affine, shift 0 e-0c114df991087062

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A038207 e-33066b18a08d7c1d

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 2]
        • preserver_image affine, shift 0 e-fbe1f1ba82c385a7

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A027465 e-3d5016cd0e3a44e4

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 3]
        • preserver_image affine, shift 0 e-f5b61f1417a02797

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A038303 e-85ae0250e4103468

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 10]
        • preserver_image affine, shift 0 e-c52c3082cda5f9b6

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A027467 e-ae6dc95f51d15092

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 15]
        • preserver_image affine, shift 0 e-6b5eff9154c3c40c

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A038255 e-b69b8b071f02daa4

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 6]
        • preserver_image affine, shift 0 e-b062da8ba0c30336

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A038243 e-bcf40186578fecd4

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 5]
        • preserver_image affine, shift 0 e-4899b779b1a74e41

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A007318 e-cb03f5f41d038dd3

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 1]
        • preserver_image affine, shift 0 e-5d14c25d8e006de4

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A038327 e-e4ffd1425a5524fc

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 12]
        • preserver_image affine, shift 0 e-bc6cf8841f41c42f

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A027466 e-ed12ae64460e2aac

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 7]
        • preserver_image affine, shift 0 e-4e6408a181c26c62

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A038231 e-f299bbf6486ea9ca

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 4]
        • preserver_image affine, shift 0 e-27282dcb1ed61456

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (expanded above)
    • basis_map A038279 e-f43b3e2d606e8300

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 8]
        • preserver_image affine, shift 0 e-7827a087c75bbe15

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+x).
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
            • ↵ theorem_check ma_wang_interval e-09c939a271fe525a (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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741