A173621 Triangle of Generalized Runyon numbers R_{n,k}^(4) read by rows.

1, 1, 4, 1, 12, 22, 1, 24, 120, 140, 1, 40, 380, 1140, 969, 1, 60, 920, 5060, 10626, 7084, 1, 84, 1890, 16380, 61425, 98280, 53820, 1, 112, 3472, 43400, 251720, 704816, 906192, 420732, 1, 144, 5880, 99960, 824670, 3518592, 7791168, 8347680, 3362260

Offset: 1

Author

R. J. Mathar, Nov 08 2010

Keywords

• easy
• nonn
• tabl

Comments

The Runyon numbers R_{n,k}^(1) are A001263, R_{n,k}^(2) are A108767. Row sums are in A002294.

Examples

			The triangle starts in row n=1 as
1;
1, 4;
1, 12, 22;
1, 24, 120, 140;
1, 40, 380, 1140, 969;
1, 60, 920, 5060, 10626, 7084;
1, 84, 1890, 16380, 61425, 98280, 53820;
1, 112, 3472, 43400, 251720, 704816, 906192, 420732;
		

References

• Chunwei Song, The Generalized Schroeder Theory, El. J. Combin. 12 (2005) #R53

Formula

T(n,k) = R(n,k,4) with R(n,k,m)= binomial(n,k)*binomial(m*n,k-1)/n, 1<=k<=n.

T(n,n) = A002293(n).

T(n,n-1) = A004332(n).

T(n,2) = A046092(n-1).