A323211 Level 1 of Pascal's pyramid. T(n, k) triangle read by rows for n >= 0 and 0 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 5, 7, 5, 2, 1, 1, 2, 6, 11, 11, 6, 2, 1, 1, 2, 7, 16, 21, 16, 7, 2, 1, 1, 2, 8, 22, 36, 36, 22, 8, 2, 1, 1, 2, 9, 29, 57, 71, 57, 29, 9, 2, 1, 1, 2, 10, 37, 85, 127, 127, 85, 37, 10, 2, 1

Offset

0,5

Comments

Pascal's pyramid is defined by recurrence. P(0) is Pascal's triangle. Now assume P(n-1) already constructed. Then P(n) is found by the steps: (1) Add 1 to each term of P(n-1). (2) Add at the left and at the right side a diagonal consisting all of 1s and complement the top with the rows 1 and 1, 1. A similar construction starting from the Pascal's triangle and subtracting 1 from all terms leads to A014473.

Links

Table of n, a(n) for n=0..77.

Formula

T(n, k) = binomial(n-2, k-1) + 1 if n != 1 else 1.

Example

Triangle starts:
                               1
                              1, 1
                            1, 2, 1
                           1, 2, 2, 1
                         1, 2, 3, 2, 1
                        1, 2, 4, 4, 2, 1
                      1, 2, 5, 7, 5, 2, 1
                    1, 2, 6, 11, 11, 6, 2, 1
                  1, 2, 7, 16, 21, 16, 7, 2, 1
                1, 2, 8, 22, 36, 36, 22, 8, 2, 1
              1, 2, 9, 29, 57, 71, 57, 29, 9, 2, 1

Maple

T := (n, k) -> `if`(n=1, 1, binomial(n-2, k-1) + 1):
seq(seq(T(n, k), k=0..n), n=0..10);
# Alternative:
T := proc(n, k) option remember;
if k = n then return 1 fi; if k < 2 then return k+1 fi;
T(n-1, k-1) + T(n-1, k) - 1 end:
seq(seq(T(n, k), k=0..n), n=0..10);

Crossrefs

Differs from A323231 only in the second term.

Row sums are A323227.

Cf. A014473.

Sequence in context: A087062 A204026 A300119 * A110537 A144434 A322057

Adjacent sequences: A323208 A323209 A323210 * A323212 A323213 A323214

Keyword

nonn,tabl,easy

Author

Peter Luschny, Feb 11 2019

Status

approved