A038321 → basis A013609

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

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 504*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 14*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-322a2e4c0bbd3231
• 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-322a2e4c0bbd3231
• 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-322a2e4c0bbd3231

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 A013609 e-322a2e4c0bbd3231

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f6f8234ea600f6f8 root: theorem check

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

    • parent_provers periodic_product e-16634379058ff4dd 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.

• A038335 → basis A013610
• A038284 → 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