cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A134146 Triangle of numbers obtained from the partition array A134145.

Original entry on oeis.org

1, 3, 1, 15, 3, 1, 105, 24, 3, 1, 945, 150, 24, 3, 1, 10395, 1485, 177, 24, 3, 1, 135135, 14805, 1620, 177, 24, 3, 1, 2027025, 191520, 16425, 1701, 177, 24, 3, 1, 34459425, 2687580, 208125, 16830, 1701, 177, 24, 3, 1, 654729075, 44552025, 2880360, 212985
Offset: 1

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Author

Wolfdieter Lang, Nov 13 2007

Keywords

Comments

This triangle is named S2(3)'.
In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

Examples

			[1]; [3,1]; [15,3,1]; [105,24,3,1]; [945,150,24,3,1];...

Links

Crossrefs

Cf. A134147 (row sums).
Cf. A134148 (allternating row sums).
Cf. A134134 (k=2 member of this triangle family).

Formula

a(n,m)=sum(product(S2(3;j,1)^e(n,m,q,j),j=1..n),q=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;j,1)= A001147(j) = A035342(j,1) = (2*j-1)!!.

A134145 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.

Original entry on oeis.org

1, 3, 1, 15, 3, 1, 105, 15, 9, 3, 1, 945, 105, 45, 15, 9, 3, 1, 10395, 945, 315, 225, 105, 45, 27, 15, 9, 3, 1, 135135, 10395, 2835, 1575, 945, 315, 225, 135, 105, 45, 27, 15, 9, 3, 1, 2027025, 135135, 31185, 14175, 11025, 10395, 2835, 1575, 945, 675, 945, 315, 225
Offset: 1

Views

Author

Wolfdieter Lang, Nov 13 2007

Keywords

Comments

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_3(3) = A134144 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(3)/M_3.

Examples

			[1]; [3,1]; [15,3,1]; [105,15,9,3,1]; [945,105,45,15,9,3,1]; ...
a(4,3)=9 from the third (k=3) partition (2^2) of 4: (3)^2 = 9, because S2(3,2,1) = 3!! = 1*3 = 3.

Links

Crossrefs

Cf. A134147 (row sums, also of triangle A134146).

Formula

a(n,k) = Product_{j=1..n} S2(3,j,1)^e(n,k,j) with S2(3,n,1) = A035342(n,1) = A001147(n) = (2*n-1)!! 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) = A134144(n,k)/A036040(n,k) (division of partition arrays M_3(3) by M_3).

A134148 Alternating row sums of triangle A134146.

Original entry on oeis.org

1, 2, 13, 83, 817, 9065, 121795, 1850384, 31964686, 612859952, 12980965588, 300358150373, 7548024748057, 204624140789585, 5954693801179750, 185127523366125449, 6124428454164394921, 214815152433502078607
Offset: 1

Views

Author

Wolfdieter Lang, Nov 13 2007

Keywords

Crossrefs

Cf. A134147 (row sums of A134146).

Formula

a(n) = Sum_{m=1..n} A134146(n,m)*(-1)^(m-1), n>=1.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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