A229032: Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

OEIS matches ?

• A015518: 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:  2;
  2:  3,   4;
  3:  4,  16;
  4:  5,  40,   16;
  5:  6,  80,   96;
  6:  7, 140,  336,   64;
  7:  8, 224,  896,  512;
  8:  9, 336, 2016, 2304,  256;
  9: 10, 480, 4032, 7680, 2560;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - 4*x)*v[i - 2]); 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, 1/4]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-44df003225948a93
• 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-e2e7a7b682b47947
• 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-e2e7a7b682b47947
• 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-e2e7a7b682b47947

Proof chains

3 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

• definition_equality e-3864cd74360cd35e root: theorem check

  Ore-algebra identity of two definitions polytool ?

• gamma_polynomial scale 1, shift 0 e-e2e7a7b682b47947

  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

  • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • preserver_image reversal, shift 0 e-75f779345b39f910

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)
• preserver_image affine, shift 0 e-44df003225948a93

  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)

Used in 7 proof steps

• e-7b35457a4f9e8cf8 preserver_image (affine, shift 0) → shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k *…
• e-85c5f3f1ee78f4c4 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1…
• e-8f28dc5012800fde preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) =…
• e-a96f03eb4914ee6b preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C…
• e-fe74f628d5055d01 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^…
• e-257278eb9a50cd75 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4…
• e-a5a16e178035b04e veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4…

Related sequences ?

Same fingerprint (2)

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

• A013609 → Veronese 2 (j=1)
• A120909 → Veronese 2 (j=1)

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [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
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023