A339019 Square table read by upwards antidiagonals: T(m,n) = A103438(2*m-1,n)/A103438(1,n) for m>=1, n>=1.

1, 1, 1, 1, 3, 1, 1, 11, 6, 1, 1, 43, 46, 10, 1, 1, 171, 386, 130, 15, 1, 1, 683, 3366, 1870, 295, 21, 1, 1, 2731, 29866, 28234, 6455, 581, 28, 1, 1, 10923, 267086, 437350, 149031, 17941, 1036, 36, 1

Offset

1,5

Links

Table of n, a(n) for n=1..45.

Formula

Let a(i,m) = ((-2)^i)*Sum_{j=0..i} C(2*m,i-j)*C(i+j,j)*((i-j)/(i+j))*B(2*m-i+j), B(s) = A027641(s)/A027642(s) the Bernoulli numbers and N = n*(n+1)/2, then T(m,n) = (1/(2*m))*Sum_{i=2..m} a(i,m)*N^(i-1)}.

Example

T(3,4) = A103438(2*3-1,4)/A103438(1,4) = 1300/10 = 130.
By formula: a(2,3) = 4*15*1*1*B(4) = -2 and a(3,3) = (-8)*15*4*(2/4)*B(4) = 8 yields T(3,n) = (-N+4*N^2)/3. Since N = 4*5/2 = 10, T(3,4) = (4*10^2-10)/3 = 130.
Table begins:
m\n| 1    2      3       4        5         6          7
---+-----------------------------------------------------
1  | 1    1      1       1        1         1          1
2  | 1    3      6      10       15        21         28
3  | 1   11     46     130      295       581       1036
4  | 1   43    386    1870     6455     17941      42868
5  | 1  171   3366   28234   149031    586341    1880956
6  | 1  683  29866  437350  3546775  19809461   85475908
7  | 1 2731 267086 6871138 85960967 683338501 3972825676

Crossrefs

Cf. A103438.

Sequence in context: A120270 A243752 A113711 * A257894 A103997 A256895

Adjacent sequences: A339016 A339017 A339018 * A339020 A339021 A339022

Keyword

nonn,tabl

Author

Franz Vrabec, Dec 24 2020

Status

approved