A038336 → basis A013610

OEIS matches ?

No OEIS matches.

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:             484,             1320,               900;
  1:           10648,            43560,             59400,              27000;
  2:          234256,          1277760,           2613600,            2376000,             810000;
  3:         5153632,         35138400,          95832000,          130680000,           89100000,            24300000;
  4:       113379904,        927653760,        3162456000,         5749920000,         5880600000,          3207600000,           729000000;
  5:      2494357888,      23809779840,       97403644800,       221371920000,       301870800000,        246985200000,        112266000000,         21870000000;
  6:     54875873536,     598645893120,     2857173580800,      7792291584000,     13282315200000,      14489798400000,       9879408000000,       3849120000000,        656100000000;
  7:   1207269217792,   14816485854720,    80817195571200,    257145622272000,    525979681920000,     717245020800000,     652040928000000,     381062880000000,     129907800000000,     19683000000000;
  8:  26559922791424,  362180765337600,  2222472878208000,   8081719557120000,  19285921670400000,   31558780915200000,   35862251040000000,   27944611200000000,   14289858000000000,   4330260000000000,    590490000000000;
  9: 584318301411328, 8764774521169920, 59759826280704000, 244472016602880000, 666741863462400000, 1272870830246400000, 1735732950336000000, 1690648977600000000, 1152715212000000000, 523961460000000000, 142898580000000000, 17714700000000000;

The row polynomials satisfy P(n,x) = (22+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1320*x + 900*x^2.

P[0, x_] := 484 + 1320*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 30*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1320*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 30*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-022a6c0efe3a8d25
• 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-022a6c0efe3a8d25
• 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-022a6c0efe3a8d25

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-022a6c0efe3a8d25

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-2f1e093ae220e593 root: theorem check

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

    • parent_provers periodic_product e-3f0a5f7ceded39cb root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A038259 → 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