A120909 → D

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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:  0;
  1:  6;
  2: 12,  24;
  3: 18,  72,   72;
  4: 24, 144,  288,   192;
  5: 30, 240,  720,   960,   480;
  6: 36, 360, 1440,  2880,  2880,  1152;
  7: 42, 504, 2520,  6720, 10080,  8064,  2688;
  8: 48, 672, 4032, 13440, 26880, 32256, 21504,  6144;
  9: 54, 864, 6048, 24192, 60480, 96768, 96768, 55296, 13824;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((n + 2*n*x)*v[i - 1])/(n - 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-13c4b00cf881ee86
• 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-13c4b00cf881ee86
• 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-13c4b00cf881ee86
• 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-13c4b00cf881ee86

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 derivative_polynomial, shift 0 e-13c4b00cf881ee86

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f5018d5f4ba5a310 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • preserver_image reversal, shift 0 e-98e1992c4c48b815

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-26e15e96a906c1df root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • parent_provers periodic_product e-3e9b67ef5f31843c root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-b63d4680864933c7

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-f5018d5f4ba5a310 (expanded above)
            • parent_provers periodic_product e-f91a02f39edc297f root: theorem check

              Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-f91a02f39edc297f (expanded above)

Related sequences ?

Same fingerprint (67)

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

• A356546 → basis A013611
• basis-map coefficients of A113214, shifted → basis A013611
• A038221 → basis A013611
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A085881 → basis A007318
• basis-map coefficients of A030528, shifted → basis A013609
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A156992 → basis A013611
• A028326 → basis A007318
• A038207 → D
• A085880 → basis A013611
• A038330 → basis A013609
• A178821 → basis A013611
• A085880 → x ↦ x+1
• A038338 → basis A007318
• A038325 → basis A013611
• A156992 → basis A007318
• basis-map coefficients of A026729 → basis A013609
• A130123 → basis A013609
• A344565 → basis A013611
• A344565 → x ↦ x/(x+1)
• A178820 → x ↦ x/(x+1)
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A196347 → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A120910 → D
• A038208 → basis A013611
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A135838 → basis A013611
• basis-map coefficients of A021012 → basis A013609
• A344565 → basis A007318
• A027466 → basis A013611
• A178821 → basis A007318
• A038243 → basis A013610
• A135838 → x ↦ x+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• basis-map coefficients of A112227 → basis A013609
• A038208 → basis A007318
• A178822 → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A085880 → x ↦ x/(x+1)
• A028326 → x ↦ x/(x+1)
• A038338 → basis A013611
• A178822 → x ↦ x+1
• A013609 → basis A013610
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A178820 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609
• A085881 → basis A013611
• A136572 → basis A013609
• A178820 → basis A007318
• A196347 → basis A007318
• A013609 → D
• A007318 → D → x ↦ x/(x+1)
• A356546 → basis A007318
• A178822 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A159854 → x ↦ x/(x+1)
• A038208 → x ↦ x/(x+1)
• A135838 → x ↦ x/(x+1)
• A038325 → basis A007318
• A120909 → basis A013610
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A178821 → x ↦ x/(x+1)
• A028326 → basis A013611

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.