A038301 → basis A013611

OEIS matches ?

• A089357: row sums, 13 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,             1760,              1936;
  1:            8000,            52800,            116160,              85184;
  2:          160000,          1408000,           4646400,            6814720,             3748096;
  3:         3200000,         35200000,         154880000,          340736000,           374809600,           164916224;
  4:        64000000,        844800000,        4646400000,        13629440000,         22488576000,         19789946880,           7256313856;
  5:      1280000000,      19712000000,      130099200000,       477030400000,       1049466880000,       1385296281600,        1015883939840,         319277809664;
  6:     25600000000,     450560000000,     3469312000000,     15264972800000,      41978675200000,      73882468352000,       81270715187200,       51084449546240,       14048223625216;
  7:    512000000000,   10137600000000,    89210880000000,    457949184000000,    1511232307200000,    3324711075840000,     4876242911232000,     4597600459161600,     2528680252538880,      618121839509504;
  8:  10240000000000,  225280000000000,  2230272000000000,  13084262400000000,   50374410240000000,  132988443033600000,   243812145561600000,   306506697277440000,   252868025253888000,   123624367901900800,   27197360938418176;
  9: 204800000000000, 4956160000000000, 54517760000000000, 359817216000000000, 1583195750400000000, 4876242911232000000, 10727734404710400000, 16857868350259200000, 18543655185285120000, 13598680469209088000, 5983419406451998720, 1196683881290399744;

The row polynomials satisfy P(n,x) = (20+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1760*x + 1936*x^2.

P[0, x_] := 400 + 1760*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 44*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 + 1760*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 44*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-9ac6ce8d07a161c9
• 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-9ac6ce8d07a161c9
• 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-9ac6ce8d07a161c9

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-9ac6ce8d07a161c9

  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 (3)

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

• A038336 → basis A007318
• A038259 → basis A007318
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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