A013612 → basis A007318

OEIS matches ?

• A001020: 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,         60,          25;
  1:       216,        540,         450,         125;
  2:      1296,       4320,        5400,        3000,         625;
  3:      7776,      32400,       54000,       45000,       18750,        3125;
  4:     46656,     233280,      486000,      540000,      337500,      112500,       15625;
  5:    279936,    1632960,     4082400,     5670000,     4725000,     2362500,      656250,       78125;
  6:   1679616,   11197440,    32659200,    54432000,    56700000,    37800000,    15750000,     3750000,      390625;
  7:  10077696,   75582720,   251942400,   489888000,   612360000,   510300000,   283500000,   101250000,    21093750,    1953125;
  8:  60466176,  503884800,  1889568000,  4199040000,  6123600000,  6123600000,  4252500000,  2025000000,   632812500,  117187500,   9765625;
  9: 362797056, 3325639680, 13856832000, 34642080000, 57736800000, 67359600000, 56133000000, 33412500000, 13921875000, 3867187500, 644531250, 48828125;

The row polynomials satisfy P(n,x) = (6+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 60*x + 25*x^2.

P[0, x_] := 36 + 60*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 5*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 + 60*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 5*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-a1d582096966c77c
• 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-a1d582096966c77c
• 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-a1d582096966c77c

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 A007318 e-a1d582096966c77c

  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 (6)

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

• A038334 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A038271 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A038220 → basis A013610
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.

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