A038284 → basis A013610

OEIS matches ?

• A009976: 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:           196,            504,             324;
  1:          2744,          10584,           13608,             5832;
  2:         38416,         197568,          381024,           326592,           104976;
  3:        537824,        3457440,         8890560,         11430720,          7348320,          1889568;
  4:       7529536,       58084992,       186701760,        320060160,        308629440,        158723712,         34012224;
  5:     105413504,      948721536,      3659354496,       7841473920,      10081895040,       7777461888,       3333197952,        612220032;
  6:    1475789056,    15179544576,     68307950592,     175649015808,     282293061120,     290358577152,     186659085312,      68568643584,      11019960576;
  7:   20661046784,   239077827072,   1229543110656,    3688629331968,    7113785140224,    9146295180288,    7839681583104,    4319824545792,    1388515032576,     198359290368;
  8:  289254654976,  3718988421120,  21517004436480,   73772586639360,  165988319938560,  256096265048064,  274388855408640,  201591812136960,   97196052280320,   27770300651520,   3570467226624;
  9: 4049565169664, 57272421685248, 368179853690880, 1420122292807680, 3651743038648320, 6573137469566976, 8451176746586112, 7761284767272960, 4989397350389760, 2138313150167040, 549851952900096, 64268410079232;

The row polynomials satisfy P(n,x) = (14+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 504*x + 324*x^2.

P[0, x_] := 196 + 504*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 18*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 504*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 18*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-89b2e37c644537dd
• 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-89b2e37c644537dd
• 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-89b2e37c644537dd

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-89b2e37c644537dd

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4ef268212a4f2f46 root: theorem check

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

    • parent_provers periodic_product e-4fdd1b3030c1c25c 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.

• A038321 → basis A013609
• A038335 → 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