A038321 → basis A013610

OEIS matches ?

• A009983: 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:            324,             756,              441;
  1:           5832,           20412,            23814,              9261;
  2:         104976,          489888,           857304,            666792,            194481;
  3:        1889568,        11022480,         25719120,          30005640,          17503290,           4084101;
  4:       34012224,       238085568,        694416240,        1080203040,         945177660,         441082908,          85766121;
  5:      612220032,      4999796928,      17499289248,       34026395760,       39697461720,       27788223204,       10806531246,        1801088541;
  6:    11019960576,    102852965376,     419982941952,      979960197888,     1429108621920,     1333834713792,      778070249712,      259356749904,       37822859361;
  7:   198359290368,   2082772548864,    9719605228032,    26458925342976,    46303119350208,    54020305908576,    42015793484448,    21007896742224,     6127303216482,      794280046581;
  8:  3570467226624,  41655450977280,  218691117630720,   680372365962240,  1389093580506240,  1944731012708736,  1890710706800160,  1260473804533440,   551457289483380,   142970408384580,   16679880978201;
  9: 64268410079232, 824777929350144, 4811204587875840, 16839216057565440, 39291504134319360, 64176123419388288, 74872143989286336, 62393453324405280, 36396181105903080, 14154070430073420, 3302616433683798, 350277500542221;

The row polynomials satisfy P(n,x) = (18+21*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 756*x + 441*x^2.

P[0, x_] := 324 + 756*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 21*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 756*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 21*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-632cc4d34c6e419b
• 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-632cc4d34c6e419b
• 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-632cc4d34c6e419b

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 A013610 e-632cc4d34c6e419b

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f6f8234ea600f6f8 root: theorem check

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

    • parent_provers periodic_product e-16634379058ff4dd 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.

• A038284 → basis A013609
• A038335 → basis A013609
• A013613 → basis A007318

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