A193973 -id:A193973 - cp's OEIS frontend

A193974 Mirror of the triangle A193973.

2, 5, 3, 9, 7, 4, 14, 12, 9, 5, 20, 18, 15, 11, 6, 27, 25, 22, 18, 13, 7, 35, 33, 30, 26, 21, 15, 8, 44, 42, 39, 35, 30, 24, 17, 9, 54, 52, 49, 45, 40, 34, 27, 19, 10, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11, 77, 75, 72, 68, 63, 57, 50, 42, 33, 23, 12, 90, 88, 85, 81

Offset: 0

Clark Kimberling, Aug 10 2011

A193974 is obtained by reversing the rows of the triangle A193973.

			First six rows:
2
5....3
9....7....4
14...12...9....5
20...18...15...11...6
27...25...22...18...13...7

Cf. A193973.

z = 13;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193973 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193974 *)

Write w(n,k) for the triangle at A193973. The triangle at A193974 is then given by w(n,n-k).

A104462 Convert the binary strings in A101305 to decimal.

Original entry on oeis.org

0, 2, 20, 328, 10512, 672800, 86118464, 22046326912, 11287719379200, 11558624644301312, 23672063271529088000, 96960771160183144450048, 794302637344220319334797312, 13013854410247705711981319168000, 426437981314996820770203866497040384

Offset: 0

Jorge Coveiro, Apr 23 2005

The a(n)-th composition in standard order is (2,3,..,n+1), where the k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. Moreover, the binary indices of a(n) are row n of A193973. Including 1 gives A164894, reverse A246534. - Gus Wiseman, Jun 28 2022

			From _Gus Wiseman_, Jun 28 2022: (Start)
The terms together with their standard compositions begin:
      0: ()
      2: (2)
     20: (2,3)
    328: (2,3,4)
  10512: (2,3,4,5)
(End)

Michael S. Branicky, Table of n, a(n) for n = 0..80

Cf. A101305.
A version for prime indices is A070826.
Cf. A000120, A002110, A029931, A066099, A070939, A164894, A193973, A233564, A246534, A272919, A333218, A333255.

convert(10,decimal,binary); convert(10100,decimal,binary); convert(101001000,decimal,binary); convert(10100100010000,decimal,binary); convert(10100100010000100000,decimal,binary);

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Range[2,n]],{n,8}] (* Gus Wiseman, Jun 28 2022 *)

def a(n): return 0 if n==0 else int("".join("1"+"0"*(i+1) for i in range(n)), 2)
print([a(n) for n in range(15)]) # Michael S. Branicky, Jun 28 2022

a(14) and beyond from Michael S. Branicky, Jun 28 2022

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A193974 Mirror of the triangle A193973.

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A104462 Convert the binary strings in A101305 to decimal.

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