basis-map coefficients of A297191, shifted → basis A008277

OEIS matches ?

• A099977: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1;
  1: 1,      3,         1;
  2: 1,     15,        25,          10,            1;
  3: 1,     63,       301,         350,          140,           21,             1;
  4: 1,    255,      3025,        7770,         6951,         2646,           462,            36,             1;
  5: 1,   1023,     28501,      145750,       246730,       179487,         63987,         11880,          1155,           55,            1;
  6: 1,   4095,    261625,     2532530,      7508501,      9321312,       5715424,       1899612,        359502,        39325,         2431,          78,          1;
  7: 1,  16383,   2375101,    42355950,    210766920,    420693273,     408741333,     216627840,      67128490,     12662650,      1479478,      106470,       4550,       105,        1;
  8: 1,  65535,  21457825,   694337290,   5652751651,  17505749898,   25708104786,   20415995028,    9528822303,   2758334150,    512060978,    62022324,    4910178,    249900,     7820,    136,     1;
  9: 1, 262143, 193448101, 11259666950, 147589284710, 693081601779, 1492924634839, 1709751003480, 1144614626805, 477297033785, 129413217791, 23466951300, 2892439160, 243577530, 13916778, 527136, 12597, 171, 1;

The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-1,x) + (3*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-7ccdf7ec12150e18
• 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-7ccdf7ec12150e18
• 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-7ccdf7ec12150e18

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-7ccdf7ec12150e18

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

  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [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