OFFSET
1,5
COMMENTS
If we define T_m(n, k) = binomial(n+m,k) - m*n where m <= k <= n, then T_0 is Pascal's triangle A007318 and T_1 is the current triangle sequence.
This modified Pascal's triangle is symmetric: C(n+m, k) - m*n = C(n+m, n-k+1) - m*n for any nonnegative integer m.
FORMULA
T(n, k) = T_1(n, k) = binomial(n+1, k) - n, for 1 <= k <= n.
EXAMPLE
The triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10
1 1;
2 1, 1;
3 1, 3, 1;
4 1, 6, 6, 1;
5 1, 10, 15, 10, 1;
6 1, 15, 29, 29, 15, 1;
7 1, 21, 49, 63, 49, 21, 1;
8 1, 28, 76, 118, 118, 76, 28, 1;
9 1, 36, 111, 201, 243, 201, 111, 36, 1;
10 1, 45, 155, 320, 452, 452, 320, 155, 45, 1; etc.
MAPLE
seq(seq(binomial(n+1, k)-n, k=1..n), n=1..10); # Muniru A Asiru, Feb 05 2018
MATHEMATICA
Table[Binomial[n + 1, k] - n, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Feb 05 2018 *)
PROG
(Magma) [[Binomial(n+1, k)- 1*n: k in [1..n]]: n in [1..10]];
(GAP) Flat(List([1..100], n->List([1..n], k->Binomial(n+1, k)-n))); # Muniru A Asiru, Feb 05 2018
(PARI) T(n, k) = binomial(n+1, k) - n;
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 01 2018
CROSSREFS
KEYWORD
AUTHOR
Juri-Stepan Gerasimov, Feb 03 2018
STATUS
approved