A299146 Modified Pascal's triangle read by rows: T(n,k) = C(n+1,k) - n, 1 <= k <= n.
1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 15, 10, 1, 1, 15, 29, 29, 15, 1, 1, 21, 49, 63, 49, 21, 1, 1, 28, 76, 118, 118, 76, 28, 1, 1, 36, 111, 201, 243, 201, 111, 36, 1, 1, 45, 155, 320, 452, 452, 320, 155, 45, 1, 1, 55, 209, 484, 781, 913, 781, 484, 209, 55, 1
Offset: 1
Examples
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.
Programs
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GAP
Flat(List([1..100],n->List([1..n],k->Binomial(n+1,k)-n))); # Muniru A Asiru, Feb 05 2018
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Magma
[[Binomial(n+1, k)- 1*n: k in [1..n]]: n in [1..10]];
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Maple
seq(seq(binomial(n+1,k)-n, k=1..n), n=1..10); # Muniru A Asiru, Feb 05 2018
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Mathematica
Table[Binomial[n + 1, k] - n, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, 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
Formula
T(n, k) = T_1(n, k) = binomial(n+1, k) - n, for 1 <= k <= n.
Comments