A196347 → basis A013610

OEIS matches ?

• A052562: 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:         2,          3;
  2:         8,         24,          18;
  3:        48,        216,         324,         162;
  4:       384,       2304,        5184,        5184,         1944;
  5:      3840,      28800,       86400,      129600,        97200,        29160;
  6:     46080,     414720,     1555200,     3110400,      3499200,      2099520,       524880;
  7:    645120,    6773760,    30481920,    76204800,    114307200,    102876480,     51438240,     11022480;
  8:  10321920,  123863040,   650280960,  1950842880,   3657830400,   4389396480,   3292047360,   1410877440,   264539520;
  9: 185794560, 2508226560, 15049359360, 52672757760, 118513704960, 177770557440, 177770557440, 114281072640, 42855402240, 7142567040;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 3*n*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] = (2*n + 3*n*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-d8646d207d983420
• 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-d8646d207d983420
• 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-d8646d207d983420

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-d8646d207d983420

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-61ffea65f7eec286 root: theorem check

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

    • basis_map A007318 e-83332bc01455e815

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers root_domain e-8128a83cc2062067 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-7d7235cf57d80f69

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)
    • parent_provers root_domain e-bf1e7898c429e3eb root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-ab4aac9d5a3bd564

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)
        • ↵ preserver_image mobius, shift 0 e-7d7235cf57d80f69 (expanded above)

Related sequences ?

Same fingerprint (23)

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

• basis-map coefficients of A113214, shifted → basis A013610
• A085880 → basis A013610
• A028326 → basis A013610
• A038243 → basis A013611
• A085881 → basis A013610
• A178821 → basis A013610
• A120910 → basis A013609
• A013620 → D
• A178822 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A356546 → basis A013610
• A038221 → basis A013610
• A038338 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A156992 → basis A013610
• A038325 → basis A013610
• A038220 → D
• A135838 → basis A013610
• A178820 → basis A013610
• A120910 → x ↦ x/(x+1)
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038208 → basis A013610
• A344565 → basis A013610

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