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.

A277132 The first subdiagonal of triangle A196842.

Original entry on oeis.org

1, 3, 14, 78, 508, 3796, 32048, 301872, 3139680, 35746080, 442258560, 5909028480, 84802072320, 1301090515200, 21253339699200, 368281038182400, 6747621163315200, 130338926671564800, 2647326900234240000, 56404832240977920000
Offset: 1

Views

Author

R. J. Mathar, Oct 01 2016

Keywords

Crossrefs

Cf. A196842.

Formula

a(n) = A196842(n,n-1).
a(n) + (-2*n-1)*a(n-1) + n^2*a(n-2) = 0.

A196843 Table of the elementary symmetric functions a_k(1,2,3,5,6...n+1) (missing 4).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 11, 41, 61, 30, 1, 17, 107, 307, 396, 180, 1, 24, 226, 1056, 2545, 2952, 1260, 1, 32, 418, 2864, 10993, 23312, 24876, 10080, 1, 41, 706, 6626, 36769, 122249, 234684, 233964, 90720, 1, 51, 1116, 13686, 103029, 489939, 1457174
Offset: 0

Views

Author

Wolfdieter Lang, Oct 25 2011

Keywords

Comments

For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x[j]=j for j=1,2,3 and x[j]=j+1 for j=4,...,n. This is the triangle S_4(n,k), n>=0, k=0..n. The first four rows coincide with those of triangle A094638.

Examples

			n\k  0   1    2    3     4      5     6      7   ...
0:   1
1:   1   1
2:   1   3    2
3:   1   6   11    6
4:   1  11   41   61    30
5:   1  17  107  307   396    180
6:   1  24  226 1056  2545   2952   1260
7:   1  32  418 2864 10993  23312  24876  10080
...
a(3,0) = a_0(1,2,3):= 1, a(3,1) = a_1(1,2,3)= 6.
a(4,2) = a_2(1,2,3,5) = 1*2+1*3+1*5+2*3+2*5+3*5 = 41.
a(4,2) = 1*|s(6,4)| - 4*|s(6,5)| + 16*|s(6,6)| =
  1*85 -4*15+16*1 = 41.

Crossrefs

Cf. A094638, A145324,|A123319|, A196841, A196842.

Formula

a(n,k) = a_k(1,2,..,n) if 0<=n<4, and a_k(1,2,3,5,...,n+1) if n>=4, with the elementary symmetric functions a_k defined in a comment to A196841.
a(n,k) = 0 if n
a(n,k)= sum((-4)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=4
with the Stirling numbers of the first kind s(n,m)=
A048994(n,m).

A196846 Table of elementary symmetric functions a_k(1,2,5,6,...,n+2) (no 3,4).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 8, 17, 10, 1, 14, 65, 112, 60, 1, 21, 163, 567, 844, 420, 1, 29, 331, 1871, 5380, 7172, 3360, 1, 38, 592, 4850, 22219, 55592, 67908, 30240, 1, 48, 972, 10770, 70719, 277782, 623828, 709320, 302400, 1, 59, 1500, 21462, 189189, 1055691, 3679430, 7571428, 8104920, 3326400
Offset: 0

Views

Author

Wolfdieter Lang, Oct 27 2011

Keywords

Comments

For the symmetric functions a_k see a comment in A196841.
The definition of the family of number triangles
S_{i,j}(n,k),n>=k>=0, 1<=i
A196845. The present triangle is S_{3,4}(n,k) (no 3 and 4
admitted). The first three lines coincide with those of
triangle A094638(n+1,k+1) which tabulates a_k(1,2,...,n).

Examples

			n\k   0    1    2     3      4      5      6       7 ...
0:    1
1:    1    1
2:    1    3    2
3:    1    8   17    10
4:    1   14   65   112     60
5:    1   21  163   567    844    420
6:    1   29  331  1871   5380   7172   3360
7:    1   38  592  4850  22219  55592  67908   30240
...
a(2,2)=a_2(1,2)=A094638(3,3)=1*2=2.
a(2,2) = |s(3,1)| = 2.
a(4,2) = a_2(1,2,5,6) = 1*2+1*5+1*6+2*5+2*6+5*6 = 65.
a(4,2) = 1*(|s(7,5)| - (3*S_3(5,1) + 4*S_4(5,1))) +
3*4*(|s(7,7)| -(3*0 + 4*0)) = 1*(175 -(3*18 + 4*17))
+ 12*1 = 65.

Crossrefs

Cf. A094638 (a_k triangle), A196845 (no 1,2 triangle), A196842 (no 3), A196843 (no 4).

Formula

a(n,k) = 0 if n=3; k=0..n, with the elementary symmetric functions a_k (see the comment above).
a(n,k) = |s(n+1,n+1-k)| for 0<=n<3,
a(n,k) = sum(((3*4)^m)*(|s(n+3,n+3-k+2*m)| - (3*S_3(n+1,k-1-2*m) + 4*S_4(n+1,k-1-2*m))),m = 0..floor(k/2)), with the Stirling numbers of the first kind s(n,m) = A048994(n,m), and the number triangles S_3(n,k)= A196842(n,k) and S_4(n,k)= A196843(n,k) (for negative k one puts the entries of these triangles to 0).

A196844 Table of the elementary symmetric functions a_k(1,2,3,4,6,...,n+1) (5 missing).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 35, 50, 24, 1, 16, 95, 260, 324, 144, 1, 23, 207, 925, 2144, 2412, 1008, 1, 31, 391, 2581, 9544, 19564, 20304, 8064, 1, 40, 670, 6100, 32773, 105460, 196380, 190800, 72576, 1, 50, 1070, 12800, 93773, 433190, 1250980
Offset: 0

Views

Author

Wolfdieter Lang, Oct 25 2011

Keywords

Comments

For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x(j) = j for j = 1, 2, 3, 4 and x(j) = j + 1 for j = 5, ..., n. This is the triangle S_5(n,k), n >= 0, k = 0..n. The first five rows coincide with those of triangle A094638.

Examples

			n\k 0   1    2     3     4      5      6     7 ...
0:  1
1:  1   1
2:  1   3    2
3:  1   6   11     6
4:  1  10   35    50    24
5:  1  16   95   260   324    144
6:  1  23  207   925  2144   2412   1008
7:  1  31  391  2581  9544  19564  20304  8064
...
a(4,0) = a_0(1, 2, 3, 4) := 1, a(4,1) = a_1(1, 2, 3, 4) = 10.
a(5,2) = a_2(1, 2, 3, 4, 6) = 1*2 + 1*3 + 1*4 + 1*6 + 2*3 + 2*4 + 2*6 + 3*4 + 3*6 + 4*6 = 95.
a(5,2) = 1*|s(7,5)| - 5*|s(7,6)| + 25*|s(7,7)| = 1*175 - 5*21 + 25*1 = 95.

Crossrefs

Cf. A094638, A145324,|A123319|, A196841, A196842, A196843.

Formula

a(n,k) = a_k(1, 2, ..., n) if 0 <= n < 5, and a_k(1, 2, 3, 4, 6, 7, ..., n+1) if n >= 5, with the elementary symmetric functions a_k defined in a comment to A196841.
a(n,k) = 0 if n < k, a(n,k) = |s(n+1, n+1-k)| if 0 <= n < 5, and
a(n,k) = sum((-5)^m*|s(n+2, n+2-k+m)|, m = 0..k) if n >= 5, with the Stirling numbers of the first kind s(n,m)=A048994(n,m).
Showing 1-4 of 4 results.

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