A038314 → basis A007318

OEIS matches ?

• A009978: 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:             484,              528,              144;
  1:           10648,            17424,             9504,              1728;
  2:          234256,           511104,           418176,            152064,             20736;
  3:         5153632,         14055360,         15333120,           8363520,           2280960,            248832;
  4:       113379904,        371061504,        505992960,         367994880,         150543360,          32845824,          2985984;
  5:      2494357888,       9523911936,      15584583168,       14167802880,        7727892480,        2529128448,        459841536,         35831808;
  6:     54875873536,     239458357248,     457147772928,      498706661376,      340027269120,      148375535616,      40466055168,       6306398208,       429981696;
  7:   1207269217792,    5926594341888,   12930751291392,    16457319825408,    13465079857152,     7344589012992,    2670759641088,     624333422592,     85136375808,      5159780352;
  8:  26559922791424,  144872306135040,  355595660513280,   517230051655680,   493719594762240,   323161916571648,  146891780259840,   45784450990080,   9365001338880,   1135151677440,    61917364224;
  9: 584318301411328, 3505909808467968, 9561572204912640, 15646209062584320, 17068591704637440, 13034197301723136, 7109562164576256, 2769959284899840, 755443441336320, 137353352970240, 14984002142208, 743008370688;

The row polynomials satisfy P(n,x) = (22+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 528*x + 144*x^2.

P[0, x_] := 484 + 528*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 12*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 + 528*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 12*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-f9a8e29d679c9a8f
• 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-f9a8e29d679c9a8f
• 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-f9a8e29d679c9a8f

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 A007318 e-f9a8e29d679c9a8f

  Geometric basis map (elementary) polytool ?

  • 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 ?

Same fingerprint (3)

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

• A038277 → basis A013610
• A013618 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.

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