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 1-5 of 5 results.

A144279 Partition number array, called M32hat(-3)= 'M32(-3)/M3'= 'A143173/A036040', related to A000369(n,m)= |S2(-3;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 21, 3, 1, 231, 21, 9, 3, 1, 3465, 231, 63, 21, 9, 3, 1, 65835, 3465, 693, 441, 231, 63, 27, 21, 9, 3, 1, 1514205, 65835, 10395, 4851, 3465, 693, 441, 189, 231, 63, 27, 21, 9, 3, 1, 40883535, 1514205, 197505, 72765, 53361, 65835, 10395, 4851, 2079, 1323, 3465
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(-3;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(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-3):= A144280(n,m).

Examples

			a(4,3) = 9 = |S2(-3,2,1)|^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

Cf. A036040, A143173, A134278, A000041, A008545.
Cf. A144274 (M32hat(-2) array), A144284 (M32hat(-4) array).

Formula

a(n,k) = Product_{j=1..n} |S2(-3,j,1)|^e(n,k,j), with |S2(-3,n,1)|= A008545(n-1) = (4*n-5)(!^4) (4-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(-3)/M3' = 'A143173/A036040' (elementwise division of arrays).

A157403 A partition product of Stirling_2 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 21, 1, 45, 84, 231, 1, 165, 840, 1155, 3465, 1, 855, 8610, 13860, 20790, 65835, 1, 3843, 64680, 250635, 291060, 460845, 1514205, 1, 21819, 689136, 3969735, 6015240, 7373520, 12113640, 40883535, 1, 114075
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 3,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143173.
Same partition product with length statistic is A000369.
Diagonal a(A000217) = A008545
Row sum is A016036.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157404, A157405

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(4*j - 1).

A144267 Partition number array, called M32(-4), related to A011801(n,m)= |S2(-4;n,m)| ( generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 36, 12, 1, 504, 144, 48, 24, 1, 9576, 2520, 1440, 360, 240, 40, 1, 229824, 57456, 30240, 12960, 7560, 8640, 960, 720, 720, 60, 1, 6664896, 1608768, 804384, 635040, 201096, 211680, 90720, 60480, 17640, 30240, 6720, 1260, 1680, 84, 1, 226606464, 53319168
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)=:M32(-4;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, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+3)-ary trees if the outdegree is r >= 0.
If M32(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle A011801(n,m)= |S2(-4;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

Examples

			a(4,3)=48. The relevant partition of 4 is (2^2). The 48 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are quaternary because r=1 vertices are quaternary (4-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4^2=16 versions due to the two quaternary root vertices.

Links

Crossrefs

Cf. A143173 (M32(-3) array), A144268 (M32(-5) array).

Formula

a(n,k) = (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-4,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-4,j,1)|^e(n,k,j),j=1..n), with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-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. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

A143171 Partition number array, called M32(-1), related to A001497(n-1,m-1) = |S2(-1;n,m)| (generalized Stirling2 triangle).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 12, 3, 6, 1, 105, 75, 30, 30, 15, 10, 1, 945, 630, 225, 90, 225, 180, 15, 60, 45, 15, 1, 10395, 6615, 2205, 1575, 2205, 1575, 630, 315, 525, 630, 105, 105, 105, 21, 1, 135135, 83160, 26460, 17640, 7875, 26460, 17640, 12600, 3150, 2520, 5880, 6300
Offset: 1

Views

Author

Wolfdieter Lang, Oct 09 2008, Dec 04 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)=:M32(-1;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, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk :=(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = Sum_{j=1..n} e(n,k,j). The special (enk)-forest is composed of m rooted increasing r-ary trees if the outdegree is r >= 0.
This generalizes the array of multinomials called M_3 in Abramowitz-Stegun, pp. 831-2. M_3 = A036040.
If M32(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle A001497(n-1,m-1) = |S2(-1;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

Examples

			a(4,3) = 3. The relevant partition of 4 is (2^2). The 3 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing unary trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are unary because r=1 vertices are unary (1-ary) and for the leaves (r=0) the arity does not matter.

Links

Crossrefs

Cf. A143173 M32(-2) array.

Formula

a(n,k) = (n!/Product_{j=1..n} e(n,k,j)!*j!^e(n,k,j)) * Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j) = M3(n,k)*Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j), with |S2(-1,n,1)| = A001147(n-1) = (2*n-3)(!^2) (2-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. Exponents 0 can be omitted due to 0!=1. M3(n,k) := A036040(n,k), k=1..p(n), p(n) := A000041(n).

A143172 Partition number array, called M32(-2), related to A004747(n,m) = |S2(-2;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 10, 6, 1, 80, 40, 12, 12, 1, 880, 400, 200, 100, 60, 20, 1, 12320, 5280, 2400, 1000, 1200, 1200, 120, 200, 180, 30, 1, 209440, 86240, 36960, 28000, 18480, 16800, 7000, 4200, 2800, 4200, 840, 350, 420, 42, 1, 4188800, 1675520, 689920, 492800, 224000, 344960
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)=:M32(-2;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, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+1)-ary trees if the outdegree is r>=0.
If M32(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle A004747(n,m)= |S2(-2;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

Examples

			a(4,3)=12. The relevant partition of 4 is (2^2). The 12 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are binary because r=1 vertices are binary (2-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4 versions due to the two binary root vertices.

Links

Crossrefs

Cf. A143171 (M32(-1) array), A143173 (M32(-3) array).

Formula

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-2,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-2,j,1)|^e(n,k,j),j=1..n), with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-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. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).
Showing 1-5 of 5 results.

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