A013612 → basis A013609

OEIS matches ?

• A001025: 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,        120,         100;
  1:       216,       1080,        1800,         1000;
  2:      1296,       8640,       21600,        24000,        10000;
  3:      7776,      64800,      216000,       360000,       300000,        100000;
  4:     46656,     466560,     1944000,      4320000,      5400000,       3600000,       1000000;
  5:    279936,    3265920,    16329600,     45360000,     75600000,      75600000,      42000000,      10000000;
  6:   1679616,   22394880,   130636800,    435456000,    907200000,    1209600000,    1008000000,     480000000,     100000000;
  7:  10077696,  151165440,  1007769600,   3919104000,   9797760000,   16329600000,   18144000000,   12960000000,    5400000000,    1000000000;
  8:  60466176, 1007769600,  7558272000,  33592320000,  97977600000,  195955200000,  272160000000,  259200000000,  162000000000,   60000000000,  10000000000;
  9: 362797056, 6651279360, 55427328000, 277136640000, 923788800000, 2155507200000, 3592512000000, 4276800000000, 3564000000000, 1980000000000, 660000000000, 100000000000;

The row polynomials satisfy P(n,x) = (6+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 120*x + 100*x^2.

P[0, x_] := 36 + 120*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 10*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 + 120*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 10*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-31086658f6c15e80
• 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-31086658f6c15e80
• 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-31086658f6c15e80

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 A013609 e-31086658f6c15e80

  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 (4)

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

• A038271 → basis A013611
• A038329 → basis A013610
• A013620 → basis A007318
• A013622: Triangle of coefficients in expansion of (3+5x)^n.

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