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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jun 06 2018, from Maple *)

CROSSREFS

Column k=0 gives A258312.

Row sums give A258308.

Cf. A258306, A258310.

Sequence in context: A097131 A192263 A109738 * A143891 A030400 A115345

Adjacent sequences:  A258304 A258305 A258306 * A258308 A258309 A258310

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, May 25 2015

STATUS

approved

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Last modified July 18 11:30 EDT 2019. Contains 325138 sequences. (Running on oeis4.)