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 1-4 of 4 results.

A144343 Row sums of triangle A144342 (called S2hat(-5)).

Original entry on oeis.org

1, 6, 61, 1021, 22801, 654271, 22642171, 922787096, 43149037646, 2279170742696, 134100508257596, 8698551285481371, 616549939781495121, 47409430650407225871, 3930455865337807339246, 349471231024984588597871, 33172028312913149756842121
Offset: 1

Views

Author

Wolfdieter Lang, Oct 09 2008

Keywords

Crossrefs

Cf. A144342.

Formula

a(n) = Sum_{m=1..n} A144342(n,m), n>=1.

A144344 Second column (m=2) of triangle S2hat(-5) = A144342.

Original entry on oeis.org

1, 5, 80, 1210, 29205, 782595, 27002800, 1058476100, 48782479625, 2522622197425, 146681902699200, 9401689444974750, 661001092169312125, 50460675421190255375, 4160330180022220820000, 368146438283724242989000, 34808031903090390296900625
Offset: 0

Views

Author

Wolfdieter Lang, Oct 09 2008

Keywords

Crossrefs

Cf. A144342, A008543 (m=1 column).

Formula

a(n) = A144342(n+2,2), n>=0.

A144285 Lower triangular array called S2hat(-4) related to partition number array A144284.

Original entry on oeis.org

1, 4, 1, 36, 4, 1, 504, 52, 4, 1, 9576, 648, 52, 4, 1, 229824, 12888, 712, 52, 4, 1, 6664896, 286272, 13464, 712, 52, 4, 1, 226606464, 8182944, 299520, 13720, 712, 52, 4, 1, 8837652096, 266366016, 8455392, 301824, 13720, 712, 52, 4, 1, 388856692224, 10191545280, 273091392
Offset: 1

Views

Author

Wolfdieter Lang Oct 09 2008

Keywords

Comments

If in the partition array M32khat(-4)= A144284 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-4). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.
The first three columns are A008546, A144339, A144340.

Examples

			[1];[4,1];[36,4,1];[504,52,4,1];[9576,648,52,4,1];...

Links

Crossrefs

Row sums A144286.
A144280 (S2hat(-3)), A144342 (S2hat(-5)).

Formula

a(n,m)=sum(product(|S2(-4;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(-4,n,1)|= A011801(n,1) = A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if n=1.

A144341 Partition number array, called M32hat(-5)= 'M32(-5)/M3'= 'A144268/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 55, 5, 1, 935, 55, 25, 5, 1, 21505, 935, 275, 55, 25, 5, 1, 623645, 21505, 4675, 3025, 935, 275, 125, 55, 25, 5, 1, 21827575, 623645, 107525, 51425, 21505, 4675, 3025, 1375, 935, 275, 125, 55, 25, 5, 1, 894930575, 21827575, 3118225, 1182775, 874225, 623645
Offset: 1

Views

Author

Wolfdieter Lang Oct 09 2008

Keywords

Comments

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(-5;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
If M32hat(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-5):= A144342(n,m).

Examples

			a(4,3)= 25 = |S2(-5,2,1)|^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

A144284 (M32hat(-4) array).

Formula

a(n,k)= product(|S2(-5,j,1)|^e(n,k,j),j=1..n) with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-factorials) for n>=2 and 1 if n=1 and 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.
Formally a(n,k)= 'M32(-5)/M3' = 'A144268/A036040' (elementwise division of arrays).
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.