A038329 → basis A013611

OEIS matches ?

• A009971: 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:           225,            360,             144;
  1:          3375,           8100,            6480,            1728;
  2:         50625,         162000,          194400,          103680,            20736;
  3:        759375,        3037500,         4860000,         3888000,          1555200,           248832;
  4:      11390625,       54675000,       109350000,       116640000,         69984000,         22394880,          2985984;
  5:     170859375,      956812500,      2296350000,      3061800000,       2449440000,       1175731200,        313528320,        35831808;
  6:    2562890625,    16402500000,     45927000000,     73483200000,      73483200000,      47029248000,      18811699200,      4299816960,       429981696;
  7:   38443359375,   276792187500,    885735000000,   1653372000000,    1984046400000,    1587237120000,     846526464000,    290237644800,     58047528960,     5159780352;
  8:  576650390625,  4613203125000,  16607531250000,  35429400000000,   49601160000000,   47617113600000,   31744742400000,  14511882240000,   4353564672000,   773967052800,    61917364224;
  9: 8649755859375, 76117851562500, 304471406250000, 730731375000000, 1169170200000000, 1309470624000000, 1047576499200000, 598615142400000, 239446056960000, 63852281856000, 10216365096960, 743008370688;

The row polynomials satisfy P(n,x) = (15+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 360*x + 144*x^2.

P[0, x_] := 225 + 360*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 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] = 225 + 360*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 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-a5d7492b5780a0c3
• 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-a5d7492b5780a0c3
• 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-a5d7492b5780a0c3

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-a5d7492b5780a0c3

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-d1ea7de7e1f61f8b root: theorem check

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

    • parent_provers periodic_product e-a6c3b06cf2d0ed0d root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (6)

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

• A038334 → basis A013609
• A013611 → basis A007318
• A038319 → basis A013611
• A013622 → basis A013609
• A038231 → x ↦ x/(x+1)
• A038271 → 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