A123319 -id:A123319 - cp's OEIS frontend

1, 1, 1, 1, 4, 3, 1, 8, 19, 12, 1, 13, 59, 107, 60, 1, 19, 137, 461, 702, 360, 1, 26, 270, 1420, 3929, 5274, 2520, 1, 34, 478, 3580, 15289, 36706, 44712, 20160, 1, 43, 784, 7882, 47509, 174307, 375066, 422568, 181440, 1, 53, 1214, 15722, 126329, 649397

Offset: 0

Wolfdieter Lang, Oct 24 2011

The elementary symmetric functions are defined by product(1-x[j]*x,j=1..n)=: sum((-1)^k*a_k(x[1],x[2],...,x[n])*x^k ,k=0..n), n>=1. Here x[1]=1 and x[j]=j+1 for j=2,..,n.
This triangle is the row reversed version of |A123319|.
In general, the triangle S_j(n,k), lists for n>=j the elementary symmetric functions
  a_k(1,2,...,j-1,j+1,...,n+1), k=0..n. For 0<=n  For j=0 one takes a_0(n,k) = a_k(1,2,...,n) which is A094638(n+1,k+1). a_1(n,k)=a_k(2,3,....,n+1)= A145324(n+1,k+1). The present triangle a(n,k) equals S_2(n,k).
  The first j rows of  the triangle S_j(n,k) coincide with the ones of triangle A094638.
  The following rows (n>=j) of S_j(n,k) are given by
  sum((-j)^m*|s(n+2,n+2-k+m)|,m=0..k), with the Stirling numbers of the first kind s(n,m) = A048994(n,m). The proof is done by iterating the obvious recurrence S_j(l,m) = a_m(1,2,...,l+1) - j*S_j(l,m-1), using a_k(1,2,...,n) =  |s(n+1,n+1-m)|, For a proof of the last equation see, e.g., the Stanley reference, p. 19, Second Proof.

			n\k  0   1   2    3     4      5      6      7  ...
0:   1
1:   1   1
2:   1   4   3
3:   1   8  19   12
4:   1  13  59  107    60
5:   1  19 137  461   702    360
6:   1  26 270 1420  3929   5274   2520
7:   1  34 478 3580 15289  36706  44712  20160
...
a(3,2) = 1*3+1*4+3*4 = 19.
a(3,2) = |s(5,3)| - 2*|s(5,4)| + 4*|s(5,5)| = 35-2*10+4*1 = 19.

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.

Cf. A094638, A145324,|A123319|.

a(n,k) = a_k(1,2,..,n) if 0<=n<2, and a_k(1,3,4,...,n+1) if n>=2, for k=0..n, with the elementary symmetric functions a_k defined above in a comment.

a(n,k) = 0 if n

= sum((-2)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=2, with the Stirling numbers of the first kind s(n,m) = A048994(n,m).

A196842 Table of the elementary symmetric functions a_k(1,2,4,5,...,n+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 7, 14, 8, 1, 12, 49, 78, 40, 1, 18, 121, 372, 508, 240, 1, 25, 247, 1219, 3112, 3796, 1680, 1, 33, 447, 3195, 12864, 28692, 32048, 13440, 1, 42, 744, 7218, 41619, 144468, 290276, 301872, 120960, 1, 52, 1164, 14658, 113799, 560658, 1734956, 3204632, 3139680, 1209600

Offset: 0

Wolfdieter Lang, Oct 24 2011

For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x[1]=1, x[2]=2, and x[j]=j+1 for j=3,...,n. This is the triangle S_3(n,k), n>=0, k=0..n. The first three rows coincide with those of triangle A094638.

			n\k   0    1    2     3      4      5     6       7  ...
0:    1
1:    1    1
2:    1    3    2
3:    1    7   14     8
4:    1   12   49    78     40
5:    1   18  121   372    508    240
6:    1   25  247  1219   3112   3796   1680
7:    1   33  447  3195  12864  28692  32048  13440
...
a(1,0) = a_0(1):= 1, a(1,1) = a_1(1)= 1.
a(3,2) = a_2(1,2,4) = 1*2 + 1*4 + 2*4 = 14.
a(3,2) = 1*|s(5,3)| - 3*|s(5,4)| + 9*|s(5,5)| = 1*35-3*10+9*1 = 14.

Cf. A094638, A145324,|A123319|, A196841, A055998 (column k=1), A002301 (diagonal), A277132 (subdiagonal).

A196842 := proc(n,k)
    if n = 1 and k =1 then
        1 ;
    else
        add( abs( combinat[stirling1](n+2,n+2-k+m))*(-3)^m,m=0..k) ;
    end if;
end proc: # R. J. Mathar, Oct 01 2016

a[n_, k_] := If[n == 1 && k == 1, 1, Sum[(-3)^m Abs[StirlingS1[n + 2, n + 2 - k + m]], {m, 0, k}]];
Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 16 2023, after R. J. Mathar *)

a(n,k) = a_k(1,2,..,n) if 0<=n<3, and a_k(1,2,4,5,...,n+1) if n>=3, with the elementary symmetric functions a_k defined in a comment to A196841.

a(n,k) = 0 if n=3, with the Stirling numbers of the first kind s(n,m)=A048994(n,m).

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

Wolfdieter Lang, Oct 25 2011

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.

			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.

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

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).

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

Wolfdieter Lang, Oct 25 2011

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.

			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.

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

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).

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A196841 Table of the elementary symmetric functions a_k(1,3,4,...,n+1).

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A196842 Table of the elementary symmetric functions a_k(1,2,4,5,...,n+1).

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

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

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