A193554 -id:A193554 - cp's OEIS frontend

A135233 Triangle A007318 * A193554, read by rows.

1, 2, 1, 5, 3, 1, 14, 7, 5, 1, 41, 15, 17, 7, 1, 122, 31, 49, 31, 9, 1, 365, 63, 129, 111, 49, 11, 1, 1094, 127, 321, 351, 209, 71, 13, 1, 3281, 255, 769, 1023, 769, 351, 97, 15, 1, 9842, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = 3^n.

Left column = A007051: (1, 2, 5, 14, 41, 122, ...).

			First few rows of the triangle:
   1;
   2,  1;
   5,  3,  1;
  14,  7,  5,  1;
  41, 15, 17,  7,  1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007051, A007318, A118801, A119258, A193554.

function T(n,k)
  if k eq n then return 1;
  elif k eq 0 then return (3^n+1)/2;
  else return (&+[(-1)^(n-k+j)*2^j*Binomial(n, j): j in [0..n-k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then (3^n_1)/2
    else add((-1)^(n-k+j)*binomial(n, j)*2^j, j=0..n-k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3^n+1)/2, Sum [(-1)^(n-k+i)* Binomial[n, i]*2^i, {i, 0, n-k}]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==n, 1, if(k==0, (3^n+1)/2, sum(j=0, n-k, (-1)^(n-k+j)*binomial(n,j)*2^j) )); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==n): return 1
    elif (k==0): return (3^n+1)/2
    else: return sum((-1)^(n-k+j)*2^j*binomial(n, j) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

Binomial transform of A193554, as infinite lower triangular matrices.

T(n,k) = Sum_{j=0..n-k} (-1)^(n-k+j)*binomial(n,j)*2^j, with T(n,n) = 1, and T(n,0) = (3^n + 1)/2. - G. C. Greubel, Nov 20 2019

Definition corrected by N. J. A. Sloane, Jul 30 2011

A193787 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=1+x^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 3, 3, 1, 8, 1, 4, 6, 4, 1, 16, 1, 5, 10, 10, 5, 1, 32, 1, 6, 15, 20, 15, 6, 1, 64, 1, 7, 21, 35, 35, 21, 7, 1, 128, 1, 8, 28, 56, 70, 56, 28, 8, 1, 256, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 512, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset: 0

Clark Kimberling, Aug 05 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. A193787 is the mirror (obtained by reversing rows) of A193554.

			First six rows:
1
1....1
1....1....2
1....2....1....4
1....3....3....1...8
1....4....6....4...1...16
(viz., Pascal's triangle with row sum at end of each row)

Cf. A193722, A193554.

z = 12; a = 1; b = 1;
p[n_, x_] := (a*x + b)^n
q[n_, x_] := 1 + x^n ; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193787 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193554 *)

cp's OEIS Frontend

A135233 Triangle A007318 * A193554, read by rows.

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