A038314 → basis A008277

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,               672,                144;
  1:           10648,             28656,              14688,                1728;
  2:          234256,           1102080,            1019520,              276480,               20736;
  3:         5153632,          40281792,           60122880,            28270080,             4769280,              248832;
  4:       113379904,        1431424512,         3249033984,          2361139200,           673090560,            77635584,             2985984;
  5:      2494357888,       50028992256,       166632657408,        175934481408,         75450009600,         14443204608,          1212309504,           35831808;
  6:     54875873536,     1730918031360,      8265450147840,      12203791810560,       7392714448896,       2089742893056,        287275548672,        18345885696,         429981696;
  7:   1207269217792,    59509723548672,    400981723176960,     807005326786560,     663935513149440,     260071490617344,      52080816291840,      5391970467840,      270888468480,       5159780352;
  8:  26559922791424,  2037817831268352,  19159275948724224,   51618089631744000,   56159549841899520,   29293088388415488,    8016454618841088,   1196518665093120,    96668484894720,    3921433067520,    61917364224;
  9: 584318301411328, 69604525336621056, 905780507616534528, 3223760510025842688, 4550585564513894400, 3075947845952864256, 1105063794832048128, 223028433925963776, 25765105198694400, 1669808117514240, 55849462530048, 743008370688;

The row polynomials satisfy P(n,x) = (22+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 484 + 672*x + 144*x^2.

P[0, x_] := 484 + 672*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 12*x)*P[n - 1, x] + 12*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 484 + 672*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 12*x)*v[i - 1] + 12*x*deriv(v[i - 1], x)); 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-047b920dffe4a1a4
• 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-047b920dffe4a1a4
• 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-047b920dffe4a1a4

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 A008277 e-047b920dffe4a1a4

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-50ef4b08e4862a18 root: theorem check

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

    • parent_provers periodic_product e-08ecf9f1e7a97c44 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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741