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.

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A145372 Partition number array, called M31hat(-5).

Original entry on oeis.org

1, 5, 1, 20, 5, 1, 60, 20, 25, 5, 1, 120, 60, 100, 20, 25, 5, 1, 120, 120, 300, 400, 60, 100, 125, 20, 25, 5, 1, 0, 120, 600, 1200, 120, 300, 400, 500, 60, 100, 125, 20, 25, 5, 1, 0, 0, 600, 2400, 3600, 120, 600, 1200, 1500, 2000, 120, 300, 400, 500, 625, 60, 100, 125, 20
Offset: 1

Views

Author

Wolfdieter Lang, Oct 17 2008

Keywords

Comments

If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3,4,5 or 6 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Fifth member (K=5) in the family M31hat(-K) of partition number arrays.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This array is array A144879 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144879/A036040'. E.g. a(4,3)= 25 = 75/3 = A144879(4,3)/A036040(4,3).
If M31hat(-5;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-5):= A145373.

Examples

			Triangle begins;
  [1];
  [5,1];
  [20,5,1];
  [60,20,25,5,1];
  [120,60,100,20,25,5,1];
  ...
a(4,3)= 25 = S1(-4;2,1)^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

Cf. A145369 (M31hat(-4)).

Formula

a(n,k) = product(S1(-5;j,1)^e(n,k,j),j=1..n) with S1(-5;n,1) = A008279(5,n-1) = [1,5,20,60,120,120,0,0,0,...], 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.

A134279 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6)/M_3.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 66, 36, 6, 1, 22176, 1056, 396, 66, 36, 6, 1, 576576, 22176, 6336, 4356, 1056, 396, 216, 66, 36, 6, 1, 17873856, 576576, 133056, 69696, 22176, 6336, 4356, 2376, 1056, 396, 216, 66, 36, 6, 1, 643458816, 17873856, 3459456, 1463616
Offset: 1

Views

Author

Wolfdieter Lang, Nov 13 2007

Keywords

Comments

Partition number array M_3(6) = A134278 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(6)/M_3.
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.

Examples

			[1]; [6,1]; [66,6,1]; [1056,66,36,6,1]; [22176,1056,396,66,36,6,1]; ...

Links

Crossrefs

Row sums give A134281 (also of triangle A134280).
Cf. A134274 (M_3(5)/M_3 partition array).

Formula

a(n,k) = Product_{j=1..n} S2(6,j,1)^e(n,k,j) with S2(6,n,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintuple- or 5-factorials) 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) = A134278(n,k)/A036040(n,k) (division of partition arrays M_3(6) by M_3).

A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 15, 3, 1, 1, 1, 105, 15, 3, 3, 1, 1, 1, 945, 105, 15, 9, 15, 3, 1, 3, 1, 1, 1, 10395, 945, 105, 45, 105, 15, 9, 3, 15, 3, 1, 3, 1, 1, 1, 135135, 10395, 945, 315, 225, 945, 105, 45, 15, 9, 105, 15, 9, 3, 1, 15, 3, 1, 3, 1, 1, 1, 2027025, 135135, 10395, 2835
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(-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,...].
If M32hat(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-1):= A144270(n,m).

Examples

			a(4,3)= 1 = |S2(-1,2,1)|^2. The relevant partition of 4 is (2^2).
[1]; [1,1]; [3,1,1]; [15,3,1,1,1]; [105,15,3,3,1,1,1]; ... [From _Wolfdieter Lang_, Oct 23 2008]

Links

Crossrefs

Cf. A144271 (M32hat(-2) array).

Formula

a(n,k)= product(|S2(-1,j,1)|^e(n,k,j),j=1..n) 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.
Formally a(n,k)= 'M32(-1)/M3' = 'A143171/A036040' (elementwise division of arrays).

Extensions

Corrected all entries. Wolfdieter Lang, Oct 23 2008

A144274 Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 10, 2, 1, 80, 10, 4, 2, 1, 880, 80, 20, 10, 4, 2, 1, 12320, 880, 160, 100, 80, 20, 8, 10, 4, 2, 1, 209440, 12320, 1760, 800, 880, 160, 100, 40, 80, 20, 8, 10, 4, 2, 1, 4188800, 209440, 24640, 8800, 6400, 12320, 1760, 800, 320, 200, 880, 160, 100, 40, 16, 80, 20
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(-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,...].
If M32hat(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-2):= A144275(n,m).

Examples

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

Links

Crossrefs

Cf. A144269 (M32hat(-1) array). A144279 (M32hat(-3) array).

Formula

a(n,k) = Product_{j=1..n} |S2(-2,j,1)|^e(n,k,j) 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.
Formally a(n,k)= 'M32(-2)/M3' = 'A143172/A036040' (elementwise division of arrays).

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).

A144353 Partition number array, called M31(3), related to A046089(n,m)= |S1(3;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 12, 9, 1, 60, 48, 27, 18, 1, 360, 300, 360, 120, 135, 30, 1, 2520, 2160, 2700, 1440, 900, 2160, 405, 240, 405, 45, 1, 20160, 17640, 22680, 25200, 7560, 18900, 10080, 11340, 2100, 7560, 2835, 420, 945, 63, 1, 181440, 161280, 211680, 241920, 126000, 70560, 181440
Offset: 1

Views

Author

Wolfdieter Lang Oct 09 2008, Oct 28 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) =: M31(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,...].
Third member (K=3) in the family M31(K) of partition number arrays.
If M31(3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(3)|:= A046089.

Examples

			[1];[3,1];[12,9,1];[60,48,27,18,1];[360,300,360,120,135,30,1];...
a(4,3)= 27 = 3*|S1(3;2,1)|^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

A049376 (row sums).
A130561 (M31(2) array), A144354 (M31(4) array).

Formula

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(3;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A001710(n+1) = (n+1)!/2!, 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. M3(n,k)=A036040.

A144356 Partition number array, called M31(6), related to A049374(n,m)= |S1(6;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 6, 1, 42, 18, 1, 336, 168, 108, 36, 1, 3024, 1680, 2520, 420, 540, 60, 1, 30240, 18144, 30240, 17640, 5040, 15120, 3240, 840, 1620, 90, 1, 332640, 211680, 381024, 493920, 63504, 211680, 123480, 158760, 11760, 52920, 22680, 1470, 3780, 126, 1, 3991680, 2661120
Offset: 1

Views

Author

Wolfdieter Lang Oct 09 2008, Oct 28 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) =: M31(6;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,...].
Sixth member (K=6) in the family M31(K) of partition number arrays.
If M31(6;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(6)|:= A049374.

Examples

			[1];[6,1];[42,18,1];[336,168,108,36,1];[3024,1680,2520,420,540,60,1];...
a(4,3)= 108 = 3*|S1(6;2,1)|^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

A049402 (row sums).
A144355 (M31(5) array).

Formula

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(6;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)|= A001725(n+4) = (n+4)!/5!, 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. M3(n,k)=A036040.

A144890 Partition number array, called M31hat(5).

Original entry on oeis.org

1, 5, 1, 30, 5, 1, 210, 30, 25, 5, 1, 1680, 210, 150, 30, 25, 5, 1, 15120, 1680, 1050, 900, 210, 150, 125, 30, 25, 5, 1, 151200, 15120, 8400, 6300, 1680, 1050, 900, 750, 210, 150, 125, 30, 25, 5, 1, 1663200, 151200, 75600, 50400, 44100, 15120, 8400, 6300, 5250, 4500
Offset: 1

Views

Author

Wolfdieter Lang Oct 09 2008, Oct 28 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) =: M31hat(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,...].
Fourth member (K=5) in the family M31hat(K) of partition number arrays.
If M31hat(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(5):= A144891.

Examples

			[1];[5,1];[30,5,1];[210,30,25,5,1];[1680,210,150,30,25,5,1];...
a(4,3)= 25 = |S1(5;2,1)|^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

Cf. A144892 (row sums).
Cf. A144885 (M31hat(4) array). A144891 (S1hat(5)).

Formula

a(n,k) = product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)| = A049353(n,1) = A001720(n+3) = [1,5,30,210,1680,...] = (n+3)!/4!, 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.

A145356 Partition number array, called M31hat(6).

Original entry on oeis.org

1, 6, 1, 42, 6, 1, 336, 42, 36, 6, 1, 3024, 336, 252, 42, 36, 6, 1, 30240, 3024, 2016, 1764, 336, 252, 216, 42, 36, 6, 1, 332640, 30240, 18144, 14112, 3024, 2016, 1764, 1512, 336, 252, 216, 42, 36, 6, 1, 3991680, 332640, 181440, 127008, 112896, 30240, 18144, 14112
Offset: 1

Views

Author

Wolfdieter Lang, Oct 17 2008, Oct 28 2008

Keywords

Comments

Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(6;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,...].
Sixth member (K=6) in the family M31hat(K) of partition number arrays.
If M31hat(6;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(6):= A145357.

Examples

			Triangle begins
  [1];
  [6,1];
  [42,6,1];
  [336,42,36,6,1];
  [3024,336,252,42,36,6,1];
  ...
a(4,3)= 36 = |S1(6;2,1)|^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

Cf. A145358 (row sums).
Cf. A144890 (M31hat(5) array), A145357 (S1hat(6)).

Formula

a(n,k) = product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)| = A049374(n,1) = A001725(n+4) = [1,6,42,336,3024,30240,332640,...] = (n+4)!/5!, 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.

A145361 Characteristic partition array for partitions with parts 1 and 2 only.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Wolfdieter Lang, Oct 17 2008

Keywords

Comments

Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to 1 if the partition has parts 1 or 2 only and to 0 otherwise.
First member (K=1) in the family M31hat(-K) of partition number arrays.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This array is array A144357 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144357/A036040'. E.g. a(4,3)= 1 = 3/3 = A144357(4,3)/A036040(4,3).
If M31hat(-1;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-1):= A145362 .

Examples

			Triangle begins:
  [1];
  [1,1];
  [0,1,1];
  [0,0,1,1,1];
  [0,0,0,0,1,1,1];
  ...
a(4,3)= 1 = S1(-1;2,1)^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

Cf. A145363 (M31hat(-2)).

Formula

a(n,k) = product(S1(-1;j,1)^e(n,k,j),j=1..n) with S1(-1;n,1) = A008279(1,n-1) = [1,1,0,0,0,...], 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.
Previous Showing 21-30 of 30 results.

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