A145324 → basis A013610

OEIS matches ?

• A051617: row sums, 15 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:
  0:        1;
  1:        3,         6;
  2:       12,        51,         54;
  3:       60,       399,        882,         648;
  4:      360,      3294,      11277,       17118,        9720;
  5:     2520,     29538,     138231,      322812,      376164,       174960;
  6:    20160,    289224,    1726146,     5485347,     9788364,      9299124,      3674160;
  7:   181440,   3086856,   22476690,    90795627,   219743604,    318612852,    256246416,     88179840;
  8:  1814400,  35767440,  308112012,  1514826900,  4648917969,   9119205828,  11165011164,   7800451632,   2380855680;
  9: 19958400, 447873840, 4462255332, 25906456260, 96582904659, 239778803178, 396391297644, 420755302872, 260202961440, 71425670400;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 2 + (3*n + 3)*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; for(n = 1, N, my(i = n + 1); v[i] = (n + 2 + (3*n + 3)*x)*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-45b4054949a52789
• 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-45b4054949a52789
• 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-45b4054949a52789
• 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-45b4054949a52789

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

• basis_map A013610 e-45b4054949a52789

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4a8bd3a5d986eb7b root: theorem check

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

    • parent_provers paper_theorem e-4000c0d63fa750fb root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-07dfe744e3acc2e8

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-419a6747cb084155 root: theorem check

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

        • preserver_image reversal, shift 0 e-986a227a8a459030

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-5f21311bb67f8c1c root: theorem check

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

            • basis_map A007318 e-0b3ba9047fae59ed

              Geometric basis map (elementary) polytool ?

              • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

                  Parent-row provers of polytool polytool ?

            • parent_provers root_domain e-2e5c10abc86c264d root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image reversal, shift 0 e-90e69a839bdbf74b

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-419a6747cb084155 (expanded above)
                • parent_provers paper_theorem e-b29f35705824bef6 root: theorem check

                  Parent-row provers of polytool polytool ?

            • preserver_image affine, shift 0 e-1ba7a8eebd893648

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers root_domain e-edbf92ee2e446027 (expanded above)
        • ↵ parent_provers paper_theorem e-b29f35705824bef6 (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