A060187 → γ

OEIS matches ?

• A333273: triangle, 41 terms agree, 11 full rows row lengths from the atlas (tabf) OEIS nonn · view at the OEIS · generated b-file 991 terms from n = 1 ?
• A182825: 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:
   1: 1;
   2: 1;
   3: 1,     4;
   4: 1,    20;
   5: 1,    72,      80;
   6: 1,   232,     976;
   7: 1,   716,    7664,     3904;
   8: 1,  2172,   49776,    88640;
   9: 1,  6544,  292320,  1217792,   354560;
  10: 1, 19664, 1618656, 13201664, 12933376;

The row polynomials satisfy P(n,x) = (1+(4*n-8)*x)*P(n-1,x) + (2*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 8)*x)*P[n - 1, x] + (2*x - 8*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 8)*x)*v[i - 1] + (2*x - 8*x^2)*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, v[i])))};
T(10)

Triangle begins:
   1: 1;
   2: 1,     4;
   3: 1,    20;
   4: 1,    72,      80;
   5: 1,   232,     976;
   6: 1,   716,    7664,      3904;
   7: 1,  2172,   49776,     88640;
   8: 1,  6544,  292320,   1217792,    354560;
   9: 1, 19664, 1618656,  13201664,  12933376;
  10: 1, 59028, 8643872, 124784768, 274820352, 51733504;

The row polynomials satisfy P(n,x) = (1+(4*n-4)*x)*P(n-1,x) + (2*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 4)*x)*P[n - 1, x] + (2*x - 8*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 4)*x)*v[i - 1] + (2*x - 8*x^2)*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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-af71d95d59cd25b7
• Coefficients of one sign. Every nonzero row, n >= 1: 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-af71d95d59cd25b7
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-af71d95d59cd25b7

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

• gamma_polynomial scale 1, shift 0 e-af71d95d59cd25b7

  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

  • premise (sequence): Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-974c6818ff00e631 root: theorem check

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

Related sequences ?

Same fingerprint (3)

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

• A257612 → γ
• A257625 → γ
• type_b_eulerian → γ

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