A038301 → basis A007318

OEIS matches ?

• A009975: 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:             400,              440,              121;
  1:            8000,            13200,             7260,             1331;
  2:          160000,           352000,           290400,           106480,            14641;
  3:         3200000,          8800000,          9680000,          5324000,          1464100,           161051;
  4:        64000000,        211200000,        290400000,        212960000,         87846000,         19326120,          1771561;
  5:      1280000000,       4928000000,       8131200000,       7453600000,       4099480000,       1352828400,        248018540,         19487171;
  6:     25600000000,     112640000000,     216832000000,     238515200000,     163979200000,      72150848000,      19841483200,       3117947360,       214358881;
  7:    512000000000,    2534400000000,    5575680000000,    7155456000000,    5903251200000,    3246788160000,    1190488992000,     280615262400,     38584598580,     2357947691;
  8:  10240000000000,   56320000000000,  139392000000000,  204441600000000,  196775040000000,  129871526400000,   59524449600000,   18707684160000,   3858459858000,   471589538200,   25937424601;
  9: 204800000000000, 1239040000000000, 3407360000000000, 5622144000000000, 6184358400000000, 4761955968000000, 2619075782400000, 1028922628800000, 282953722920000, 51874849202000, 5706233412220, 285311670611;

The row polynomials satisfy P(n,x) = (20+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 440*x + 121*x^2.

P[0, x_] := 400 + 440*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 400 + 440*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 11*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-337d052427983e0b
• 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-337d052427983e0b
• 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-337d052427983e0b

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-337d052427983e0b

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c85fc2223c79991e root: theorem check

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

    • parent_provers periodic_product e-38b35fa9e37eb25d 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.

• A038336 → basis A013611
• A038259 → basis A013611

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