A013612 → basis A013611

OEIS matches ?

• A009970: 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:        36,         240,          400;
  1:       216,        2160,         7200,          8000;
  2:      1296,       17280,        86400,        192000,         160000;
  3:      7776,      129600,       864000,       2880000,        4800000,        3200000;
  4:     46656,      933120,      7776000,      34560000,       86400000,      115200000,        64000000;
  5:    279936,     6531840,     65318400,     362880000,     1209600000,     2419200000,      2688000000,      1280000000;
  6:   1679616,    44789760,    522547200,    3483648000,    14515200000,    38707200000,     64512000000,     61440000000,     25600000000;
  7:  10077696,   302330880,   4031078400,   31352832000,   156764160000,   522547200000,   1161216000000,   1658880000000,   1382400000000,     512000000000;
  8:  60466176,  2015539200,  30233088000,  268738560000,  1567641600000,  6270566400000,  17418240000000,  33177600000000,  41472000000000,   30720000000000,  10240000000000;
  9: 362797056, 13302558720, 221709312000, 2217093120000, 14780620800000, 68976230400000, 229920768000000, 547430400000000, 912384000000000, 1013760000000000, 675840000000000, 204800000000000;

The row polynomials satisfy P(n,x) = (6+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 240*x + 400*x^2.

P[0, x_] := 36 + 240*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 20*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 240*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 20*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-db3287eb465d1cf9
• 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-db3287eb465d1cf9
• 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-db3287eb465d1cf9

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 A013611 e-db3287eb465d1cf9

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of coefficients in expansion of (1+5x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-8bdb70660e92b5b3 root: theorem check

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

    • parent_provers periodic_product e-41a50f5538a6b055 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• A038291 → 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