login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A258310 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258309(n,i); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows. 4

%I #12 May 01 2022 13:37:15

%S 1,1,2,1,4,3,9,14,3,21,50,15,51,204,122,15,127,784,644,105,323,3212,

%T 4115,1310,105,835,13068,22587,9270,945,2188,55475,137503,85109,16764,

%U 945,5798,238073,787127,614779,149754,10395

%N T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258309(n,i); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

%H Alois P. Heinz, <a href="/A258310/b258310.txt">Rows n = 0..200, flattened</a>

%e Triangle T(n,k) begins:

%e : 1;

%e : 1;

%e : 2, 1;

%e : 4, 3;

%e : 9, 14, 3;

%e : 21, 50, 15;

%e : 51, 204, 122, 15;

%e : 127, 784, 644, 105;

%e : 323, 3212, 4115, 1310, 105;

%e : 835, 13068, 22587, 9270, 945;

%e : 2188, 55475, 137503, 85109, 16764, 945;

%p b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,

%p `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)

%p +b(x-1, y, false, k) +b(x-1, y+1, true, k)))

%p end:

%p A:= (n, k)-> b(n, 0, false, k):

%p T:= proc(n, k) option remember;

%p add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!

%p end:

%p seq(seq(T(n, k), k=0..n/2), n=0..14);

%t b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0,

%t If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1]

%t + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];

%t A[n_, k_] := b[n, 0, False, k];

%t T[n_, k_] := Sum[A[n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}]/k!;

%t Table[Table[T[n, k], {k, 0, n/2}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, May 01 2022, after _Alois P. Heinz_ *)

%Y Column k=0 gives A001006.

%Y T(2n,n) gives A001147.

%Y Row sums give A258311.

%Y Cf. A258307, A258309.

%K nonn,tabf

%O 0,3

%A _Alois P. Heinz_, May 25 2015

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 07:18 EDT 2024. Contains 370954 sequences. (Running on oeis4.)