A038330 → basis A008277

OEIS matches ?

No OEIS matches.

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:            256,             144,              16;
  1:           4096,            3904,             960,              64;
  2:          65536,           94464,           38656,            5632,             256;
  3:        1048576,         2151424,         1305600,          312320,           30720,            1024;
  4:       16777216,        47222784,        39940096,        13967360,         2232320,          159744,           4096;
  5:      268435456,      1011564544,      1147453440,       550846464,       127303680,        14680064,         802816,         16384;
  6:     4294967296,     21305032704,     31585140736,     20013514752,      6277103616,      1037697024,       90832896,       3932160,        65536;
  7:    68719476736,    443280523264,    843263508480,    686718976000,    280921374720,     62465507328,     7784103936,     536346624,     18874368,      262144;
  8:  1099511627776,   9140488372224,  22011446296576,  22601185361920,  11736359895040,   3372443762688,   561226186752,   54735667200,   3051356160,    89128960,   1048576;
  9: 17592186044416, 187207813955584, 564836664606720, 720878975320064, 465968258088960, 168353415036928, 35938822520832, 4653274103808, 365407764480, 16840130560, 415236096, 4194304;

The row polynomials satisfy P(n,x) = (16+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 256 + 144*x + 16*x^2.

P[0, x_] := 256 + 144*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 144*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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-1e482bc7f83a739a
• 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-1e482bc7f83a739a
• 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-1e482bc7f83a739a

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 A008277 e-1e482bc7f83a739a

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-016885d52a801213 root: theorem check

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

    • parent_provers periodic_product e-0a35d6a32dc0856a 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.

• A027465 → basis A008277
• A143495 → D+1

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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741