A156365: T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.

OEIS matches ?

• A000670: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,    8,       4;
  1: 1,   22,      44,        8;
  2: 1,   52,     264,      208,         16;
  3: 1,  114,    1208,     2416,        912,         32;
  4: 1,  240,    4764,    19328,      19056,       3840,          64;
  5: 1,  494,   17172,   124952,     249904,     137376,       15808,        128;
  6: 1, 1004,   58432,   705872,    2499040,    2823488,      934912,      64256,        256;
  7: 1, 2026,  191360,  3641536,   20965664,   41931328,    29132288,    6123520,     259328,       512;
  8: 1, 4072,  610548, 17627904,  155809824,  503175936,   623239296,  282046464,   39075072,   1042432,    1024;
  9: 1, 8166, 1913084, 81501480, 1061095584, 5200393152, 10400786304, 8488764672, 2608047360, 244874752, 4180992, 2048;

The row polynomials satisfy P(n,x) = (1+(2*n+4)*x)*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 4*x^2.

P[0, x_] := 1 + 8*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n + 4)*x)*P[n - 1, x] + (x - 2*x^2)*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 + 8*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (2*n + 4)*x)*v[i - 1] + (x - 2*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 - 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-4e3e568f497fe445
• 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-4e3e568f497fe445
• 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-4e3e568f497fe445
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-4e3e568f497fe445

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

• theorem_check liu_wang e-4e3e568f497fe445 root: theorem check

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

Used in 2 proof steps

• e-00ad498f5c132fdf obreschkoff_combination → obreschkoff_combination of T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A1…
• e-00ad498f5c132fdf obreschkoff_combination → obreschkoff_combination of T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A1…

Related sequences ?

Same fingerprint (1)

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

• A142075: Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.

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