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

OEIS matches ?

• A142075: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A000670: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A034172: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,    2;
   3: 1,    8,      4;
   4: 1,   22,     44,       8;
   5: 1,   52,    264,     208,       16;
   6: 1,  114,   1208,    2416,      912,       32;
   7: 1,  240,   4764,   19328,    19056,     3840,       64;
   8: 1,  494,  17172,  124952,   249904,   137376,    15808,     128;
   9: 1, 1004,  58432,  705872,  2499040,  2823488,   934912,   64256,    256;
  10: 1, 2026, 191360, 3641536, 20965664, 41931328, 29132288, 6123520, 259328, 512;

The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-2*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 + (2*n - 2)*x)*P[n - 1, x] + (x - 2*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 + (2*n - 2)*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, v[i])))};
T(10)

Triangle begins:
   1: 1;
   2: 1,    2;
   3: 1,    8,      4;
   4: 1,   22,     44,       8;
   5: 1,   52,    264,     208,       16;
   6: 1,  114,   1208,    2416,      912,       32;
   7: 1,  240,   4764,   19328,    19056,     3840,       64;
   8: 1,  494,  17172,  124952,   249904,   137376,    15808,     128;
   9: 1, 1004,  58432,  705872,  2499040,  2823488,   934912,   64256,    256;
  10: 1, 2026, 191360, 3641536, 20965664, 41931328, 29132288, 6123520, 259328, 512;

The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-2*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 + (2*n - 2)*x)*P[n - 1, x] + (x - 2*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (2*n - 2)*x)*v[i - 1] + (x - 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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-5f2e9a40e80d3367
• 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-5f2e9a40e80d3367
• 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-5f2e9a40e80d3367
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-5f2e9a40e80d3367

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-5f2e9a40e80d3367 root: theorem check

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

Used in 2 proof steps

• e-111ed7b99403026c obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eu…
• e-111ed7b99403026c obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eu…

Related sequences ?

Same fingerprint (1)

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

• 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…

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