A356546 → 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:        1;
  1:        4,         2;
  2:       24,        30,          6;
  3:      160,       380,        180,         20;
  4:     1120,      4550,       3850,        980,         70;
  5:     8064,     53172,      71820,      31500,       5040,        252;
  6:    59136,    614460,    1248324,     840840,     226380,      24948,       924;
  7:   439296,   7066488,   20828808,   20252232,    8168160,    1489488,    120120,     3432;
  8:  3294720,  81145350,  338931450,  457837380,  259729470,   69729660,   9189180,   566280,   12870;
  9: 24893440, 932094020, 5428179900, 9928447100, 7616809200, 2825162340, 541140600, 53968200, 2625480, 48620;

The row polynomials satisfy n*P(n,x) = (8*n-4+(4*n-2)*x)*P(n-1,x) + (4*n-2)*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[Together[((8*n - 4 + (4*n - 2)*x)*P[n - 1, x] + (4*n - 2)*x*D[P[n - 1, x], x])/(n)]];
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] = ((8*n - 4 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*x*deriv(v[i - 1], x))/(n)); 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-398f569c764d9a1d
• 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-398f569c764d9a1d
• 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-398f569c764d9a1d

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-398f569c764d9a1d

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

  • premise (sequence): Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-883c9c594ff729de root: theorem check

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

Related sequences ?

Same fingerprint (19)

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

• A038325 → basis A008277
• A028326 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A178821 → basis A008277
• A038208 → basis A008277
• A085881 → basis A008277
• A038221 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A135838 → basis A008277
• A178822 → basis A008277
• A126351 → reverse
• A156992 → basis A008277
• A143494 → reverse
• A344565 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A085880 → basis A008277
• A196347 → basis A008277
• A038338 → basis A008277
• A178820 → 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