A147716 → basis A013610

OEIS matches ?

• A001027: 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:          15,           3;
  2:         225,          90,           9;
  3:        3375,        2025,         405,          27;
  4:       50625,       40500,       12150,        1620,         81;
  5:      759375,      759375,      303750,       60750,       6075,        243;
  6:    11390625,    13668750,     6834375,     1822500,     273375,      21870,       729;
  7:   170859375,   239203125,   143521875,    47840625,    9568125,    1148175,     76545,     2187;
  8:  2562890625,  4100625000,  2870437500,  1148175000,  287043750,   45927000,   4592700,   262440,   6561;
  9: 38443359375, 69198046875, 55358437500, 25833937500, 7750181250, 1550036250, 206671500, 17714700, 885735, 19683;

The row polynomials satisfy P(n,x) = (15+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[(15 + 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] = (15 + 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-011a72a323f5e83e
• 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-011a72a323f5e83e
• 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-011a72a323f5e83e

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-011a72a323f5e83e

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of coefficients in expansion of (14 + x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-08b3ad2096e0d7df root: theorem check

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

    • parent_provers periodic_product e-7403f1e58efb6a95 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (5)

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

• A038291 → basis A013609
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
• Triangle of coefficients in expansion of (1+5x)^n.
• A013611 → x ↦ x/(x+1)
• A038329 → basis A007318

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