|
|
A110541
|
|
A number triangle of sums of binomial products.
|
|
2
|
|
|
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 7, 4, 1, 1, 1, 8, 19, 13, 5, 1, 1, 1, 13, 51, 46, 21, 6, 1, 1, 1, 21, 141, 166, 89, 31, 7, 1, 1, 1, 34, 393, 610, 393, 151, 43, 8, 1, 1, 1, 55, 1107, 2269, 1761, 776, 235, 57, 9, 1, 1, 1, 89, 3139, 8518, 7985, 4056, 1363, 344, 73, 10, 1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,8
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = Sum_{j=0..n-k} C((k-1)*(n-k)-(k-2)*j, j)*C(j, n-k-j).
|
|
EXAMPLE
|
Rows begin
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 3, 3, 1, 1;
1, 5, 7, 4, 1, 1;
1, 8, 19, 13, 5, 1, 1;
1, 13, 51, 46, 21, 6, 1, 1;
1, 21, 141, 166, 89, 31, 7, 1, 1;
As a number square read by antidiagonals, rows begin
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 3, 7, 13, 21, 31, ...
1, 5, 19, 46, 89, 151, ...
1, 8, 51, 166, 393, 776, ...
|
|
MATHEMATICA
|
T[n_, k_] := Sum[Binomial[(k-1)*(n-k) - (k-2)*j, j]*Binomial[j, n-k-j], {j, 0, n-k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 31 2017 *)
|
|
PROG
|
(PARI) for(n=0, 20, for(k=0, n, print1(sum(j=0, n-k, binomial((k-1)*(n-k) -(k-2)*j, j)*binomial(j, n-k-j)), ", "))) \\ G. C. Greubel, Aug 31 2017
(Magma) [[(&+[Binomial((k-1)*(n-k)-(k-2)*j, j)*Binomial(j, n-k-j): j in [0..n-k]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 19 2019
(Sage) [[sum(binomial((k-1)*(n-k) -(k-2)*j, j)*binomial(j, n-k-j) for j in (0..n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 19 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j-> Binomial((k-1)*(n-k)-(k-2)*j, j)*Binomial(j, n-k-j) )))); # G. C. Greubel, Feb 19 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|