A156992 → basis A008277

OEIS matches ?

• A138379: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:          4,           2;
  2:         24,          30,            6;
  3:        192,         456,          216,           24;
  4:       1920,        7800,         6600,         1680,          120;
  5:      23040,      151920,       205200,        90000,        14400,          720;
  6:     322560,     3351600,      6809040,      4586400,      1234800,       136080,        5040;
  7:    5160960,    83018880,    244702080,    237928320,     95961600,     17498880,     1411200,      40320;
  8:   92897280,  2287958400,   9556444800,  12909093120,   7323281280,   1966083840,   259096320,   15966720,    362880;
  9: 1857945600, 69567724800, 405137376000, 741019104000, 568487808000, 210858681600, 40388544000, 4027968000, 195955200, 3628800;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 2 + (n + 1)*x)*P[n - 1, x] + (n + 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 2 + (n + 1)*x)*v[i - 1] + (n + 1)*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-d06f83defd222c3e
• 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-d06f83defd222c3e
• 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-d06f83defd222c3e

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-d06f83defd222c3e

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

  • premise (sequence): Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-d07da0cb5a6ac36b root: theorem check

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

    • parent_provers root_domain e-e811f0fabc9e8e88 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (19)

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

• A085880 → basis A008277
• A344565 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A178820 → basis A008277
• A038208 → basis A008277
• A196347 → basis A008277
• A178821 → basis A008277
• A135838 → basis A008277
• A038325 → basis A008277
• A028326 → basis A008277
• A356546 → basis A008277
• A038221 → basis A008277
• A085881 → basis A008277
• A178822 → basis A008277
• A143494 → reverse
• A126351 → reverse
• A038338 → basis A008277

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