Alford Arnold - cp's OEIS frontend

A182779 a(n) = A049019(n) * A118851(n). Irregular table read by rows.

1, 1, 2, 2, 3, 12, 6, 4, 24, 24, 72, 24, 5, 40, 120, 180, 360, 480, 120, 6, 60, 240, 180, 360, 2160, 720, 1440, 4320, 3600, 720, 7, 84, 420, 840, 630, 5040, 3780, 7560, 3360, 30240, 20160, 12600, 50400, 30240, 5040

Offset: 0

Alford Arnold, Dec 01 2010

The sequences have shape A000041 and their respective row sums are A000670, A006906 and A006153.

			For n = 3 the values are (3,12,6) = (1,6,6)*(3,2,1).
Table starts:
1;
1;
2, 2;
3, 12, 6;
4, (24, 24), 72, 24;
5, (40, 120), (180, 360), 480, 120;
6, (60, 240, 180), (360, 2160, 720), (1440, 4320), 3600, 720;
7, (84, 420, 840), (630, 5040, 3780, 7560), (3360, 30240, 20160), (12600, 50400), 30240, 5040;

Cf. A006153 (related to function composition), A133314 (signed version of A049019).

a(n) = A049019(n) * A118851(n).

a(0) = 1 prepended by Peter Luschny, May 31 2020

A181417 Irregular triangle T(n,k) = binomial(n-1,m-1)m!A036040(n,k), where m=A036043(n,k).

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 24, 18, 108, 24, 1, 40, 80, 360, 540, 960, 120, 1, 60, 150, 100, 900, 3600, 900, 4800, 10800, 9000, 720, 1, 84, 252, 420, 1890, 9450, 6300, 9450, 16800, 100800, 50400, 63000, 189000, 90720, 5040, 1, 112, 392, 784, 490, 3528, 21168, 35280, 26460, 35280, 47040, 352800, 235200, 705600, 88200, 294000, 2352000, 1764000, 846720, 3175200, 987840, 40320, 1, 144, 576, 1344, 2016, 6048, 42336

Offset: 1

Alford Arnold, Oct 22 2010

This is a refinement of the triangle A048743.
Row n has A000041(n) elements.
The sequence can be derived by expanding A007318 and A000142 and using A036040.
For example, row four can be derived using
(1 3 3 3 1) times (1 2 2 6 24) times (1 4 3 6 1) = (1 24 18 108 24)

			The table begins:
1
1...2
1..12...6
1..24..18..108..24

Cf. A045531 (row sums), A048743, A007318, A036040.

Row 6 onwards and definition by R. J. Mathar, Feb 12 2013

A181512 Irregular triangle a(n,k) = A181415(n,k) / A036040(n,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 3, 6, 6, 1, 4, 4, 12, 12, 24, 24, 1, 5, 5, 5, 20, 20, 20, 60, 60, 120, 120, 1, 6, 6, 6, 30, 30, 30, 30, 120, 120, 120, 360, 360, 720, 720, 1, 7, 7, 7, 7, 42, 42, 42, 42, 42, 210, 210

Offset: 1

Alford Arnold, Oct 26 2010

This is an irregular table related to labeled rooted trees and to Bell numbers.

A181511 contains the same values, without repetition.

			1;
1,1;
1,2,2;
1,3,3,6,6;
1,4,4,12,12,24,24;
1,5,5,5,20,20,20,60,60,120,120;
1,6,6,6,30,30,30,30,120,120,120,360,360,720,720;

Cf. A181511, A181513 (row sums).

sum_{k=1.. A000041(n)} a(n,k) = A181513(n). (Row sums)

A181513 Row sums of A181512.

Original entry on oeis.org

1, 2, 5, 19, 81, 436, 2659

Offset: 1

Alford Arnold, Oct 26 2010

nonn

			a(4) = 1+3+3+6+6= 19.

Cf. A181511, A181512

A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

Original entry on oeis.org

1, 1, 1, 1, 6, 2, 1, 12, 9, 36, 6, 1, 20, 40, 120, 180, 240, 24, 1, 30, 75, 50, 300, 1200, 300, 1200, 2700, 1800, 120, 1, 42, 126, 210, 630, 3150, 2100, 3150, 4200, 25200, 12600, 12600, 37800, 15120, 720, 1, 56, 196, 392, 245, 1176, 7056, 11760, 8820, 11760, 11760, 88200

Offset: 1

Alford Arnold, Oct 20 2010

			Row three is calculated as follows:
( 3 18 6) divided by (3 3 3) yielding (1 6 2)
1;
1,1;
1,6,2;
1,12,9,36,6;
1,20,40,120,180,240,24;
1,30,75,50,300,1200,300,1200,2700,1800,120;
1,42,126,210,630,3150,2100,3150,4200,25200,12600,12600,37800,15120,720;

Cf. A000169 (row sums), A000081 (unlabeled rooted trees) A179438 (a similar refinement), A054589, A135278, A019538, A101817, A101818

Sum_{k=1.. A000041(n)} a(n,k) = A000169(n). (Row sums)

a(n,k) = A098546(n,k) *A049019(n,k) /n. - Compare with the formula in A101818.

Edited by R. J. Mathar, May 17 2016

A181511 Triangle T(n,k) = n!/(n-k)! read by rows, 0 <= k < n.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 12, 24, 1, 5, 20, 60, 120, 1, 6, 30, 120, 360, 720, 1, 7, 42, 210, 840, 2520, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 1, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800

Offset: 1

Alford Arnold, Oct 26 2010

Row n contains the same set of values as row A181512(n,.), which are related to labeled rooted trees (A000169) and Bell numbers (A000110) respectively.

			The triangle begins:
  1;
  1,  2;
  1,  3,  6;
  1,  4, 12, 24;
which is A181512 without duplicates.

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Cf. A002627 (row sums).

a181511 n k = a181511_tabl !! (n-1) !! k
a181511_row n = a181511_tabl !! (n-1)
a181511_tabl = tail $ map init a008279_tabl
-- Reinhard Zumkeller, Nov 18 2012

A181511 := proc(n,k) n!/(n-k)! ; end proc:
seq(seq(A181511(n,k),k=0..n-1),n=1..16) ; # R. J. Mathar, Mar 03 2011

T(n,k) = A008279(n,k). - R. J. Mathar, Mar 03 2011

A181416 Irregular table T(n,k) = n*A178883(n,k) read by rows.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 32, 16, 72, 96, 120, 120, 120, 180, 180, 480, 600, 720, 576, 576, 288, 648, 1296, 216, 1152, 1728, 3600, 4320, 5040, 3360, 3360, 3360, 3024, 6048, 3024, 3024, 4032, 12096, 4032, 8400, 16800, 30240, 35280, 40320

Offset: 1

Alford Arnold, Oct 17 2010

The row sum of row n is A001286(n).

			In row n=3 the products are (3,3,3) times (2,4,6) yielding (6,12,18) which adds to 36, the third Lah number.
The table starts in row n=1 with row lengths A000041(n) as:
1;
2,4;
6,12,18;
24,32,16,72,96;
120,120,120,180,180,480,600;

Cf. A051683.

T(n,k) = A036042(n,k)*A178883(n,k), 1<=k<= A000041(n).

A179864 The unrestricted partition statistic defined by A049085(n,k)+A036043(n,k)- 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 4, 5, 5, 6, 6, 5, 4, 6, 5, 4, 6, 5, 6, 6, 7, 7, 6, 5, 7, 6, 5, 5, 7, 6, 5, 7, 6, 7, 7, 8, 8, 7, 6, 5, 8, 7, 6, 6, 5, 8, 7, 6, 6, 5, 8, 7, 6, 8, 7, 8, 8, 9, 9, 8, 7, 6, 9, 8, 7, 6, 7, 6, 5, 9, 8, 7, 7, 6, 6, 9, 8, 7, 7, 6, 9, 8, 7, 9, 8, 9, 9, 10, 10, 9, 8, 7, 6, 10, 9, 8

Offset: 1

Alford Arnold, Aug 02 2010

			1;
2,2;
3,3,3;
4,4,3,4,4;
5,5,4,5,4,5,5;
6,6,5,4,6,5,4,6,5,6,6;
7,7,6,5,7,6,5,5,7,6,5,7,6,7,7;

Cf. A000041 (row lengths), A179862 (row sums), A105805 (the rank statistic)

A179862 An unrestricted partition statistic: sum of A179864 over row n.

Original entry on oeis.org

1, 4, 9, 19, 33, 59, 93, 150, 226, 342, 494, 721, 1011, 1425, 1960, 2695, 3633, 4903, 6506, 8633, 11312, 14796, 19157, 24773, 31744, 40608, 51578, 65372, 82341, 103522, 129428, 161505, 200589, 248614, 306869, 378051, 463987, 568387, 693989, 845754, 1027625

Offset: 1

Alford Arnold, Aug 02 2010

nonn

Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n minus the number of partitions of n. - Omar E. Pol, Jul 15 2013

Sum of the hook-lengths of the (1,1)-cells of the Ferrers diagrams over all partitions of n. Example: a(3) = 9 because in each of the partitions 3, 21, and 111 the (1,1)-cell has hook-length 3. Comment follows at once from the previous comment. - Emeric Deutsch, Dec 20 2015

			From _Omar E. Pol_, Jul 15 2013: (Start)
Illustration of initial terms using a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). a(n) is the x-coordinate of the mentioned largest peak. Note that this Dyck path is infinite.
.
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
. 0,2,  6,   12,         24,             40... = A211978
.  1, 4,   9,       19,           33... = this sequence (End)

Cf. A179864.

a(n) = Sum_{k=1..A000041(n)} A179864(n,k).
a(n) = A211978(n) - A000041(n). - Omar E. Pol, Jul 15 2013
a(n) = A225600(A139582(n)-1), n>= 1. - Omar E. Pol, Jul 25 2013

More terms from Omar E. Pol, Jul 15 2013

A179974 Triangle read by rows: T(n,k) = (n-A049085(n,k))! in columns 1<=k<=A000041(n), rows n>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 6, 1, 1, 2, 2, 6, 6, 24, 1, 1, 2, 6, 2, 6, 24, 6, 24, 24, 120, 1, 1, 2, 6, 2, 6, 24, 24, 6, 24, 120, 24, 120, 120, 720, 1, 1, 2, 6, 24, 2, 6, 24, 24, 120, 6, 24, 120, 120, 720, 24, 120, 720, 120, 720, 720, 5040, 1, 1, 2, 6, 24, 2, 6, 24, 120, 24, 120, 720, 6, 24, 120, 120, 720, 720, 24, 120, 720, 720, 5040, 120, 720, 5040

Offset: 1

Alford Arnold, Aug 05 2010

Since A049085 is a resortment of A036043 both A179972 and A179974 have row sums equal to A179973.

			Triangle begins
1;
1,1;
1,1,2;
1,1,2,2,6;
1,1,2,2,6,6,24;
1,1,2,6,2,6,24,6,24,24,120;
1,1,2,6,2,6,24,24,6,24,120,24,120,120,720;
1,1,2,6,24,2,6,24,24,120,6,24,120,120,720,24,120,720,120,720,720,5040;
1,1,2,6,24,2,6,24,120,24,120,720,6,24,120,120,720,720,24,120,720,720,5040,120,720,5040,720,5040,5040,40320,

Cf. A000041 (row lengths), A179973 (row sums), A036042, A049085 (max part).

cp's OEIS Frontend

User: Alford Arnold

A182779 a(n) = A049019(n) * A118851(n). Irregular table read by rows.

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A181417 Irregular triangle T(n,k) = binomial(n-1,m-1)*m!*A036040(n,k), where m=A036043(n,k).

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A181512 Irregular triangle a(n,k) = A181415(n,k) / A036040(n,k).

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A181513 Row sums of A181512.

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A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

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A181511 Triangle T(n,k) = n!/(n-k)! read by rows, 0 <= k < n.

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A181416 Irregular table T(n,k) = n*A178883(n,k) read by rows.

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A179864 The unrestricted partition statistic defined by A049085(n,k)+A036043(n,k)- 1.

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A179862 An unrestricted partition statistic: sum of A179864 over row n.

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A179974 Triangle read by rows: T(n,k) = (n-A049085(n,k))! in columns 1<=k<=A000041(n), rows n>=1.

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A181417 Irregular triangle T(n,k) = binomial(n-1,m-1)m!A036040(n,k), where m=A036043(n,k).