A256595 Triangle A074909(n) with 0's as second column.

1, 1, 0, 1, 0, 3, 1, 0, 6, 4, 1, 0, 10, 10, 5, 1, 0, 15, 20, 15, 6, 1, 0, 21, 35, 35, 21, 7, 1, 0, 28, 56, 70, 56, 28, 8, 1, 0, 36, 84, 126, 126, 84, 36, 9, 1, 0, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset

0,6

Comments

For Bernoulli numbers, B(1) excluded.

B(n) is calculated via

B(0) = 1;

B(0) + 0 = 1;

B(0) + 0 + 3*B(2) = 3/2;

B(0) + 0 + 6*B(2) + 4*B(3) = 2;

etc.

The diagonal is A026741(n+1)/A040001(n).

Row sums: 1, 1, 4, 11, 26, 57, ..., essentially Euler numbers A000295. See A130103, A008292 and A173018.

There is an infinitude of Bernoulli number sequences. They are of the form

B(n,q) = 1, q, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, ... .

Chronologically, the first, and the most regular, is, for q=1/2, A164555(n)/A027642(n), from Jacob Bernoulli (1654-1705), published in Ars Conjectandi in 1713 and(?) Seko Kowa (1642-1708) in 1712. See A159688. The second is, for q=-1/2, B(n,-1/2) = A027641(n)/A027642(n), from B(n,1/2) via Pascal's triangle. We could choose Be(n,q) instead of B(n,q) to avoid confusion with Sloane's B(n,p) for A027641(n)/A027642(n) (p=-1), A164555(n)/A027642(n) (p=1), A164558(n)/A027642(n) (p=2), A157809(n)/A027642(n) (p=3), ..., successive binomial transforms of the previous sequence.

This motivates the proposal of the (independent of q) sequence Bernoulli(n+2):

B(n+2) = 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ... and its inverse binomial transform. See A190339.

References

Jacob Bernoulli, Ars Conjectandi (1713).

Links

Table of n, a(n) for n=0..65.

Wikipedia, Seki Takakazu (also known as Seki Kowa).

Example

1,
1, 0,
1, 0,  3,
1, 0,  6, 4,
1, 0, 10, 10,  5,
1, 0, 15, 20, 15, 6,
1, 0, 21, 35, 35, 21, 7,
etc.

Mathematica

T[_, 0] = 1; T[_, 1] = 0; T[n_, k_] := Binomial[n+1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 11 2016 *)

Crossrefs

Cf. A007318, A074909, A026741, A004001, A000295, A130103, A008292, A173018, A027641/A027642, A164555/A027642, A176327/A176289.

Sequence in context: A308639 A348191 A213668 * A004590 A206294 A058150

Adjacent sequences: A256592 A256593 A256594 * A256596 A256597 A256598

Keyword

nonn,tabl

Author

Paul Curtz, Apr 03 2015

Status

approved