A128099 → basis A013609

OEIS matches ?

• A015441: 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:   1;
  1:   1;
  2:   3,    4;
  3:   5,    8;
  4:  11,   28,   16;
  5:  21,   64,   48;
  6:  43,  164,  192,   64;
  7:  85,  376,  544,  256;
  8: 171,  876, 1584, 1152,  256;
  9: 341, 1968, 4176, 3840, 1280;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (2 + 4*x)*v[i - 2] + 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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-2ee1895757bd17cf, e-51a31f555c517eab
• 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-2ee1895757bd17cf, e-51a31f555c517eab
• 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-2ee1895757bd17cf, e-51a31f555c517eab
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-2ee1895757bd17cf

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A013609 e-2ee1895757bd17cf

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 0 e-250eeed7ad07d148

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
        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-1f8dc4b132ba6116 root: theorem check

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

        • preserver_image reversal, shift 0 e-8efc9aafb30c2198

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
            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-1f8dc4b132ba6116 (expanded above)
    • preserver_image reversal, shift 0 e-6b9d711da9b2e815

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): A Jacobsthal triangle.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • basis_map A027467 e-0db30c1d149f8e9a

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
            • preserver_image affine, shift 0 e-0ffa0091fcab81ab

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check

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

        • basis_map A038303 e-1078e1fe9c978d23

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
            • preserver_image affine, shift 0 e-ba5819a43d999a58

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A038327 e-39068ae2f1d878f6

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
            • preserver_image affine, shift 0 e-0ef3f7736abad08b

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • preserver_image reversal, shift 0 e-5066005127b21217

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ gamma_polynomial scale 1, shift 0 e-250eeed7ad07d148 (expanded above)
            • parent_provers paper_theorem e-7c3667820056c57c root: theorem check

              Parent-row provers of polytool polytool ?

        • basis_map A027465 e-76bf35985f9d5eb4

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
            • preserver_image affine, shift 0 e-5f9e6216c74dfca3

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A038243 e-7b55c4767302dd2a

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
            • preserver_image affine, shift 0 e-0c61b86c2e25257b

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A007318 e-89ed187733702f47

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
            • preserver_image affine, shift 0 e-38de1ebb8f2e73b6

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A038255 e-9c1dd42dfa910ea7

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
            • preserver_image affine, shift 0 e-4867241b67312c9e

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A038207 e-a0ae771232cbba16

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
            • preserver_image affine, shift 0 e-c345fe1d5254381d

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A038291 e-a2d6a4286b164d3d

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
            • preserver_image affine, shift 0 e-f1a9ca5f4baaa1dd

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A038231 e-b4c854eb781c5167

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
            • preserver_image affine, shift 0 e-b26782bfa4993a06

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A038279 e-c41010d0d9363851

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
            • preserver_image affine, shift 0 e-765910d978fe3ce9

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A027466 e-de853701eca80777

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of A Jacobsthal triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
            • preserver_image affine, shift 0 e-2f0380ac472e9edc

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): A Jacobsthal triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
                • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
        • basis_map A113953 e-4e53ceff1d4735da

          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
        • ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    • ↵ parent_provers paper_theorem e-7c3667820056c57c (expanded above)
• basis_map A013611 e-51a31f555c517eab

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image affine, shift 0 e-61c0454a07890647

      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)
    • preserver_image affine, shift 0 e-ddc26a351d976abe

      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 (expanded above)
        • ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (expanded above)
    • basis_map A007318 e-2a29a8b9938e2bd7

      Geometric basis map (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 (expanded above)
        • ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (expanded above)
    • basis_map A007318 e-d0c14addeb793323

      Geometric basis map (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 (expanded above)
        • ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [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
2. [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