A038336 → basis A013611

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,              1760,               1600;
  1:           10648,             58080,             105600,              64000;
  2:          234256,           1703680,            4646400,            5632000,             2560000;
  3:         5153632,          46851200,          170368000,          309760000,           281600000,           102400000;
  4:       113379904,        1236871680,         5622144000,        13629440000,         18585600000,         13516800000,          4096000000;
  5:      2494357888,       31746373120,       173162035200,       524733440000,        954060800000,       1040793600000,        630784000000,         163840000000;
  6:     54875873536,      798194524160,      5079419699200,     18470617088000,      41978675200000,      61059891200000,      55508992000000,       28835840000000,        6553600000000;
  7:   1207269217792,    19755314472960,    143675014348800,    609530363904000,    1662355537920000,    3022464614400000,    3663593472000000,     2854748160000000,     1297612800000000,     262144000000000;
  8:  26559922791424,   482907687116800,   3951062894592000,  19156668579840000,   60953036390400000,  132988443033600000,  201497640960000000,   209348198400000000,   142737408000000000,   57671680000000000,   10485760000000000;
  9: 584318301411328, 11686366028226560, 106239691165696000, 579489224540160000, 2107233543782400000, 5363867202355200000, 9752485822464000000, 12665566003200000000, 11514150912000000000, 6978273280000000000, 2537553920000000000, 419430400000000000;

The row polynomials satisfy P(n,x) = (22+40*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1760*x + 1600*x^2.

P[0, x_] := 484 + 1760*x + 1600*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 40*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 + 1760*x + 1600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 40*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-865f1fe93a5cf324
• 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-865f1fe93a5cf324
• 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-865f1fe93a5cf324

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-865f1fe93a5cf324

  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 (2)

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

• A038301 → basis A007318
• A038259 → basis A013611

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