A134826 -id:A134826 - cp's OEIS frontend

A134285 Triangle of numbers obtained from the partition array A134284.

1, 3, 1, 10, 3, 1, 35, 19, 3, 1, 126, 65, 19, 3, 1, 462, 331, 92, 19, 3, 1, 1716, 1190, 421, 92, 19, 3, 1, 6435, 5587, 1805, 502, 92, 19, 3, 1, 24310, 20613, 8771, 2075, 502, 92, 19, 3, 1, 92378, 92821, 35726, 10616, 2318, 502, 92, 19, 3, 1, 352716, 347930, 160205

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is called s2(3)'.

			Triangle starts:
1
3, 1
10, 3, 1
35, 19, 3, 1
126, 65, 19, 3, 1
...
a(4,2)=19 because the m=2 parts partitions (1^1,3^1) and (2^2) of n=4 lead to 1^1*10^1 + 3^2 =19, since A001700(n-1)=[1,3,10,...], n>=1.

Wolfdieter Lang, First 10 rows and more.

Row sums A134826. Alternating row sums A134827.

Cf. A001700.

a(n,m) = sum(product(s2(3;j,1)^e(n,m,q,j),j=1..n),k=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. s2(3;n,1) = A035324(n,1) = A001700(n-1) = binomial(2*n-1,n).

Row sums = A001700. Triangle A134285 = A001263 * A000012. - Gary W. Adamson, Nov 19 2007

A134284 A certain partition array in Abramowitz-Stegun order (A-St order), called M_0(3)/M_0.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 35, 10, 9, 3, 1, 126, 35, 30, 10, 9, 3, 1, 462, 126, 105, 100, 35, 30, 27, 10, 9, 3, 1, 1716, 462, 378, 350, 126, 105, 100, 90, 35, 30, 27, 10, 9, 3, 1, 6435, 1716, 1386, 1260, 1225, 462, 378, 350, 315, 300, 126, 105, 100, 90, 81, 35, 30, 27, 10, 9, 3, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_0(3)= A134283 with each entry divided by the corresponding one of the partition number array M_0 = M_0(2) = A048996; in short M_0(3)/M_0.

			[1]; [3,1]; [10,3,1]; [35,10,9,3,1]; [126,35,30,10,9,3,1]; ...
a(4,3) = 9 = 3^2 because (2^2) is the k=4 partition of n=4 in A-St order and s2(3,2,1)=3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134826 (row sums coinciding with those of triangle A134285).

a(n,k) = Product_{j=1..n} s2(3,j,1)^e(n,k,j) with s2(3,n,1) = A035324(n,1) = A001700(n-1) = binomial(2*n-1,n) and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134283(n,k)/A048996(n,k) (division of partition arrays M_0(3) by M_0).

A134827 Alternating row sums of triangle A134285, called s2(3)'.

Original entry on oeis.org

1, 2, 8, 18, 78, 206, 872, 2226, 10820, 26558, 129248, 335886, 1630174, 3990978, 21778450, 48749804, 274937594, 650637382, 3598954596, 7610277626, 49741540498, 93757794966, 639655810986, 1233712869392, 8530089042832

Offset: 1

Wolfdieter Lang, Nov 13 2007

Cf. A134826 (row sums of triangle A134285).

a(n) = Sum_{m=1..n} A134285(n,m)*(-1)^(m-1).

cp's OEIS Frontend

A134285 Triangle of numbers obtained from the partition array A134284.

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A134284 A certain partition array in Abramowitz-Stegun order (A-St order), called M_0(3)/M_0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A134827 Alternating row sums of triangle A134285, called s2(3)'.

Views

Author

Keywords

Crossrefs

Formula