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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A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 66, 1, 144, 264, 1056, 1, 600, 4620, 5280, 22176, 1, 4950, 68640, 110880, 133056, 576576, 1, 26586, 639870, 3141600, 3259872, 4036032, 17873856, 1, 234528, 10759056, 69263040, 105557760, 113008896, 142990848
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 = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134278.
Same partition product with length statistic is A049385.
Diagonal a(A000217) = A008548.
Row sum is A049412.

Links

Crossrefs

Cf. A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, 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}(-5*j - 1).

Extensions

Offset corrected by Peter Luschny, Mar 14 2009

A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 45, 1, 105, 180, 585, 1, 425, 2700, 2925, 9945, 1, 3075, 34650, 52650, 59670, 208845, 1, 15855, 308700, 1248975, 1253070, 1461915, 5221125, 1, 123515, 4475520, 23689575, 33972120, 35085960, 41769000
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 = -5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134273.
Same partition product with length statistic is A049029.
Diagonal a(A000217) = A007696.
Row sum is A049120.

Links

Crossrefs

Cf. A157396, A157398, A157399, A157400, A080510, A157401, A157402, A157403, 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).

Extensions

Offset corrected by Peter Luschny, Mar 14 2009

A157398 A partition product of Stirling_2 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 28, 1, 72, 112, 280, 1, 280, 1400, 1400, 3640, 1, 1740, 15120, 21000, 21840, 58240, 1, 8484, 126420, 401800, 382200, 407680, 1106560, 1, 57232, 1538208, 6370000, 8357440, 8153600, 8852480, 24344320, 1
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134149.
Same partition product with length statistic is A035469.
Diagonal a(A000217) = A007559.
Row sum is A049119.

Links

Crossrefs

Cf. A157396, A157397, A157399, A157400, A080510, A157401, A157402, A157403, 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}(-3*j - 1).

A157399 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, 15, 1, 45, 60, 105, 1, 165, 600, 525, 945, 1, 855, 5250, 6300, 5670, 10395, 1, 3843, 39900, 91875, 79380, 72765, 135135, 1, 21819, 391440, 1164975, 1323000, 1164240, 1081080, 2027025, 1
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 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 A134144.
Same partition product with length statistic is A035342.
Diagonal a(A000217) = A001147.
Row sum is A049118.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157400, A080510, A157401, A157402, A157403, 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}(-2*j - 1).

A157401 A partition product of Stirling_2 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 9, 12, 15, 1, 25, 60, 75, 105, 1, 75, 330, 450, 630, 945, 1, 231, 1680, 3675, 4410, 6615, 10395, 1, 763, 9408, 30975, 41160, 52920, 83160, 135135, 1, 2619, 56952, 233415, 489510, 555660, 748440, 1216215
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 1,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143171.
Same partition product with length statistic is A001497.
Diagonal a(A000217) = A001147.
Row sum is A001515.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157402, A157403, 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}(2*j - 1).

A157402 A partition product of Stirling_2 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 2, 1, 6, 10, 1, 24, 40, 80, 1, 80, 300, 400, 880, 1, 330, 2400, 3600, 5280, 12320, 1, 1302, 15750, 47600, 55440, 86240, 209440, 1, 5936, 129360, 588000, 837760, 1034880, 1675520, 4188800, 1, 26784, 1146040, 5856480
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 2,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143172.
Same partition product with length statistic is A004747.
Diagonal a(A000217) = A008544.
Row sum is A015735.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157403, 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}(3*j - 1).

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).

A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 36, 1, 72, 144, 504, 1, 280, 1800, 2520, 9576, 1, 1740, 22320, 37800, 57456, 229824, 1, 8484, 182700, 864360, 1005480, 1608768, 6664896, 1, 57232, 2380896, 16546320, 26276544, 32175360, 53319168, 226606464
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 = 4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144267.
Same partition product with length statistic is A011801.
Diagonal a(A000217) = A008546.
Row sum is A028575.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, 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}(5*j - 1).

A157405 A partition product of Stirling_2 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 55, 1, 105, 220, 935, 1, 425, 3300, 4675, 21505, 1, 3075, 47850, 84150, 129030, 623645, 1, 15855, 415800, 2323475, 2709630, 4365515, 415800, 2323475, 2709630, 4365515, 21827575, 1, 123515, 6394080, 51934575
Offset: 0

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144268.
Same partition product with length statistic is A013988.
Diagonal a(A000217) = A008543.
Row sum is A028844.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404

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}(6*j - 1).

A157385 A partition product of Stirling_1 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 30, 1, 105, 120, 210, 1, 425, 1800, 1050, 1680, 1, 3075, 18600, 18900, 10080, 15120, 1, 15855, 174300, 338100, 211680, 105840, 151200, 1, 123515, 2227680, 4865700, 4327680, 2540160, 1209600, 1663200, 1, 757755
Offset: 1

Views

Author

Peter Luschny, Mar 07 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144355.
Same partition product with length statistic is A049353.
Diagonal a(A000217(n)) = rising_factorial(5,n-1), A001720(n+3).
Row sum is A049378.

Links

Crossrefs

Cf. A157386, A157384, A157383, A157400, A126074, A157391, A157392, A157393, A157394, A157395

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-2}(j-n-3).
Previous Showing 11-20 of 25 results. Next

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