A258307 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258306(n,i); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

1, 1, 2, 1, 5, 2, 14, 9, 1, 43, 28, 3, 141, 114, 21, 1, 490, 421, 82, 4, 1785, 1750, 442, 38, 1, 6789, 7114, 1941, 180, 5, 26809, 30854, 9868, 1210, 60, 1, 109632, 134239, 46337, 6191, 335, 6, 462755, 609276, 235035, 37321, 2700, 87, 1, 2012441, 2800134, 1157603, 199424, 15806, 560, 7

Offset

0,3

Links

Alois P. Heinz, Rows n = 0..200, flattened

Example

Triangle T(n,k) begins:
:     1;
:     1;
:     2,     1;
:     5,     2;
:    14,     9,    1;
:    43,    28,    3;
:   141,   114,   21,    1;
:   490,   421,   82,    4;
:  1785,  1750,  442,   38,  1;
:  6789,  7114, 1941,  180,  5;
: 26809, 30854, 9868, 1210, 60, 1;

Maple

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
                  +b(x-1, y, false, k) +b(x-1, y+1, true, k)))
    end:
A:= (n, k)-> b(n, 0, false, k):
T:= proc(n, k) option remember;
       add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
    end:
seq(seq(T(n, k), k=0..n/2), n=0..13);

Mathematica

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (x + k*y)/y, 1] + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];
A[n_, k_] :=  b[n, 0, False, k];
T[n_, k_] := T[n, k] = Sum[A[n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}]/ k!;
Table[T[n, k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jun 06 2018, from Maple *)

Crossrefs

Column k=0 gives A258312.

Row sums give A258308.

Cf. A258306, A258310.

Sequence in context: A364487 A192263 A109738 * A343254 A143891 A030400

Adjacent sequences: A258304 A258305 A258306 * A258308 A258309 A258310

Keyword

nonn,tabf

Author

Alois P. Heinz, May 25 2015

Status

approved