A038305 → basis A013611

OEIS matches ?

• A009969: 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:           169,            312,            144;
  1:          2197,           6084,           5616,            1728;
  2:         28561,         105456,         146016,           89856,           20736;
  3:        371293,        1713660,        3163680,         2920320,         1347840,          248832;
  4:       4826809,       26733096,       61691760,        75928320,        52565760,        19408896,         2985984;
  5:      62748517,      405451956,     1122790032,      1727369280,      1594494720,       883104768,       271724544,        35831808;
  6:     815730721,     6023857632,    19461693888,     35929281024,     41456862720,     30614298624,     14129676288,      3726508032,       429981696;
  7:   10604499373,    88098917868,   325288312128,    700620979968,    970090587648,    895468234752,    551057375232,    218000719872,     50307858432,     5159780352;
  8:  137858491849,  1272539924760,  5285935072080,  13011532485120,  21018629399040,  23282174103552,  17909364695040,   9446697861120,   3270010798080,   670771445760,   61917364224;
  9: 1792160394037, 18197320924068, 83987635034160, 232581143171520, 429380572008960, 554891816134656, 512207830278144, 337719448535040, 155870514708480, 47960158371840, 8854183084032, 743008370688;

The row polynomials satisfy P(n,x) = (13+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 312*x + 144*x^2.

P[0, x_] := 169 + 312*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(13 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 312*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (13 + 12*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-7b93be005ec5b750
• 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-7b93be005ec5b750
• 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-7b93be005ec5b750

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-7b93be005ec5b750

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e186153c671b61f0 root: theorem check

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

    • parent_provers periodic_product e-f733aee5869de636 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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