A063967 → Veronese 2 (j=1)

OEIS matches ?

• A155020: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A028859: row sums, 14 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:
   1:   0;
   2:   1;
   3:   3;
   4:   7,    1;
   5:  15,    7;
   6:  30,   29,    1;
   7:  58,   95,   11;
   8: 109,  271,   67,   1;
   9: 201,  705,  303,  15;
  10: 365, 1717, 1140, 121, 1;

The row polynomials satisfy P(n,x) = 2*P(n-1,x) + (1+x)*P(n-2,x) - (2-2*x)*P(n-3,x) - (1-x)*P(n-4,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 3, P(4,x) = 7 + x.

P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 3;
P[4, x_] := 7 + x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] + (1 + x)*P[n - 2, x] - (2 - 2*x)*P[n - 3, x] - (1 - x)*P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 3; v[4] = 7 + x; for(n = 5, N, my(i = n); v[i] = 2*v[i - 1] + (1 + x)*v[i - 2] - (2 - 2*x)*v[i - 3] - (1 - 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-32612196073bf5c5
• 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-32612196073bf5c5
• 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-32612196073bf5c5

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=1, shift 0 e-32612196073bf5c5

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

  • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A063967 e-6e2f4b29fae24343

      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
    • basis_map A027466 e-07199d0e633dc35b

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 7]
        • preserver_image affine, shift 0 e-12ad66072da87843

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • theorem_check ma_wang_interval e-32a5b77539bbfdf5 root: theorem check

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

    • basis_map A038291 e-19f9843f81ec4d82

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 9]
        • preserver_image affine, shift 0 e-61f403043a781326

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A038255 e-20a66a93533ac8e0

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 6]
        • preserver_image affine, shift 0 e-cc7a67409dceb768

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A038279 e-2d6f22aa44a34472

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 8]
        • preserver_image affine, shift 0 e-f5bf3eab46ece5d6

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A027465 e-300fe6b47fa78df1

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 3]
        • preserver_image affine, shift 0 e-acc0c42734fdb8fc

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A038327 e-37edf5cceba94f47

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 12]
        • preserver_image affine, shift 0 e-4d81c1e2745a9ed0

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A007318 e-50014fa936b4943d

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image reversal, shift -1 e-24564bef34b2c2e5

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check

              Parent-row provers of polytool polytool ?

            • parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check

              Parent-row provers of polytool polytool ?

            • gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25

              Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

              • premise (sequence): delannoy
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check

                  Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

                • preserver_image reversal, shift 0 e-266d85ca03a298fa

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): delannoy
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
                    • basis_map A008288 e-e771e35dc9ee3fbd

                      Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

                      • 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 [0, 0]
                        no recorded property covers this requirement
                    • basis_map A008288 e-33dba7e79fe9caa8

                      Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

                      • 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 [0, 0]
                        no recorded property covers this requirement
                • ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
                • ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)
        • basis_map A007318 e-fcd19baf8f835cf0

          Geometric basis map (elementary) polytool ?

          • 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 (-inf, 1]
            • preserver_image affine, shift 0 e-1365676e8cb0fd13

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
                • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
            • preserver_image affine, shift 0 e-3c4e27dedb9c54a8

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]
                • theorem_check ma_wang_interval e-0178a3d1cd265fb6 root: theorem check

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

        • basis_map A026729 e-fa68694d112454ed

          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-0178a3d1cd265fb6 (expanded above)
    • basis_map A038231 e-5c88965e8c99a752

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 4]
        • preserver_image affine, shift 0 e-0e7024e7b55599e9

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A038303 e-78cb35d9024a5ff1

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 10]
        • preserver_image affine, shift 0 e-25462190c1ff20a9

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A007318 e-933c6ae38be5f965

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ preserver_image reversal, shift -1 e-24564bef34b2c2e5 (expanded above)
        • ↵ basis_map A007318 e-fcd19baf8f835cf0 (expanded above)
        • ↵ basis_map A026729 e-fa68694d112454ed (expanded above)
        • ↵ theorem_check ma_wang_interval e-0178a3d1cd265fb6 (expanded above)
    • basis_map A038243 e-a054677672f9106d

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 5]
        • preserver_image affine, shift 0 e-81aeef92852c73bb

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • basis_map A038207 e-a7cf73c813009d2a

      Geometric basis map (elementary) polytool ?

      • 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 (-inf, 2]
        • ↵ preserver_image affine, shift 0 e-1365676e8cb0fd13 (expanded above)
        • ↵ preserver_image affine, shift 0 e-3c4e27dedb9c54a8 (expanded above)
    • basis_map A027467 e-bfdacf28446aab8d

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 15]
        • preserver_image affine, shift 0 e-79b40901050ca651

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
            • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    • preserver_image affine, shift 0 e-e504746edac4d52e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ preserver_image reversal, shift -1 e-24564bef34b2c2e5 (expanded above)
        • ↵ basis_map A007318 e-fcd19baf8f835cf0 (expanded above)
        • ↵ basis_map A026729 e-fa68694d112454ed (expanded above)
        • ↵ theorem_check ma_wang_interval e-0178a3d1cd265fb6 (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. [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
5. [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