A094638 -id:A094638 - cp's OEIS frontend

A199220 Triangle read by rows: T(n,k) = (n-1-k)*abs(s(n,n+1-k)), where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

-1, 0, -1, 1, 0, -2, 2, 6, 0, -6, 3, 20, 35, 0, -24, 4, 45, 170, 225, 0, -120, 5, 84, 525, 1470, 1624, 0, -720, 6, 140, 1288, 5880, 13538, 13132, 0, -5040, 7, 216, 2730, 18144, 67347, 134568, 118124, 0, -40320, 8, 315, 5220, 47250, 253092, 807975, 1447360, 1172700, 0, -362880, 9, 440, 9240, 108900, 788865, 3608220, 10250790, 16819000, 12753576, 0, -3628800

Offset: 1

Frank M Jackson, Nov 04 2011

Use the T(n,k) as coefficients to generate a polynomial of degree n-1 in d as Sum_{k=1..n} T(n,k)d^(k-1) and let g(n) be the greatest root of this polynomial. Then a polygon of n sides that form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : ... : 1/(1+(n-1)d) can only exist if the common difference d of the denominators is limited to the range f(n) < d < g(n). The lower limit f(n) is the greatest root of another group of polynomials defined by coefficients in the triangle A199221.

			Triangle starts:
  -1;
   0, -1;
   1,  0,  -2;
   2,  6,   0,  -6;
   3, 20,  35,   0, -24;
   4, 45, 170, 225,   0, -120;

Cf. A094638, A192918, A199221.

Flatten[Table[(n-1-k)Abs[StirlingS1[n,n+1-k]],{n,1,20},{k,1,n}]]

T(n,k) = (n-1-k)*abs(stirling(n,n+1-k,1)); \\ Michel Marcus, Sep 30 2018

The triangle of coefficients can be generated by expanding the equation (Sum_{k=1..n} 1/(1+(k-1)*d)) - 2 = 0 into a polynomial of degree n-1 in d.

A199221 Triangle read by rows: T(n,k) = (n+1-k)|s(n,n+1-k)| - 2|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

Original entry on oeis.org

-1, 0, 1, 1, 4, 2, 2, 12, 18, 6, 3, 28, 83, 88, 24, 4, 55, 270, 575, 500, 120, 5, 96, 705, 2490, 4324, 3288, 720, 6, 154, 1582, 8330, 23828, 35868, 24696, 5040, 7, 232, 3178, 23296, 98707, 242872, 328236, 209088, 40320, 8, 333, 5868, 57078, 334740, 1212057, 2658472, 3298932, 1972512, 362880, 9, 460, 10140, 126300, 977865, 4873680, 15637290, 31292600, 36207576, 20531520, 3628800

Offset: 1

Frank M Jackson, Nov 04 2011

Use the T(n,k) as coefficients to generate a polynomial of degree n-1 in d as Sum_{k=1..n} T(n,k)d^(k-1) and let f(n) be the greatest root of this polynomial. Then a polygon of n sides that form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : ... : 1/(1+(n-1)d) can only exist if the common difference d of the denominators is limited to the range f(n) < d < g(n). The higher limit g(n) is the greatest root of another group of polynomials defined by coefficients in the triangle A199220.

			Triangle starts:
  -1;
   0,  1;
   1,  4,   2;
   2, 12,  18,   6;
   3, 28,  83,  88,  24;
   4, 55, 270, 575, 500, 120;

Cf. A094638, A192918, A199220.

Flatten[Table[(n+1-k)Abs[StirlingS1[n,n+1-k]]-2Abs[StirlingS1[n-1,n-k]],{n,1,20},{k,1,n}]]

T(n,k) = (n+1-k)*abs(stirling(n,n+1-k,1)) - 2*abs(stirling(n-1,n-k,1));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Sep 30 2018

The triangle of coefficients can be generated by expanding the equation (Sum_{k=1..n} 1/(1+(k-1)d)) - 2/(1+(n-1)d) = 0 into a polynomial of degree n-1 in d.

A218234 Infinitesimal generator for padded Pascal matrix A097805 (as lower triangular matrices).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0

Offset: 0

Tom Copeland, Oct 24 2012

Matrix T begins
0;
0,0;
0,1,0;
0,0,2,0;
0,0,0,3,0;
0,0,0,0,4,0;
Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
Then M(1) = the lower triangular padded Pascal matrix A097805, with inverse M(-1).
Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x), the matrix T represents the action of R^2*L in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x) = x^n/n!, L = DxD and R = D^(-1).
See A132440 for an analog and more general discussion.

P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
T. Copeland, Mathemagical Forests
T. Copeland, Addendum to Mathemagical Forests
G. Dattoli, B. Germano, M. Martinelli, and P. Ricci, Touchard like polynomials and generalized Stirling polynomials
W. Lang, Combinatorial interpretation of generalized Stirling numbers

Table[PadLeft[{n-1, 0}, n+1], {n, 0, 12}]  // Flatten (* Jean-François Alcover, Apr 30 2014 *)

The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x):
  1) b(0) = 0, b(1) = 0, b(n) = (n-1) * a(n-1),
  2) B(x) = x^2D A(x)= x (xDx)(1/x)A(x) = x^2 * Lag(1,-:xD:) A(x)/x , or
  3) EB(x) = D^(-1)xD  EA(x),
  where D is the derivative w.r.t. x, (D^(-1)x^j/j!) = x^(j+1)/(j+1)!, (:xD:)^j = x^j*D^j, and Lag(n,x) are the Laguerre polynomials A021009.
So the exponentiated operator can be characterized as
  4) exp(t*T) A(x) = exp(t*x^2D) A(x) = x exp(t*xDx)(1/x)A(x)
     = x [sum(n=0,1,...) (t*x)^n * Lag(n,-:xD:)] A(x)/x
     = x [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x)/x (eval. at u=x)
     = A[x/(1-t*x)], a special Moebius or linear fractional trf.,
  5) exp(t*T) EA(x) =  D^(-1) exp(t*x)D EA(x), a shifted Euler trf.
     for an e.g.f., or
  6) [exp(t*T) * a]_n = [M(t) * a]_n
     = [sum(k=0,...,n-1) binomial(n-1,k)*  t^(n-1-k) * a(k+1)] with [M(t) * a]_0 = a_0
For generalizations and more on the operator x^2D, see A132440 and the references therein and above, and A094638.

A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; nx+1, nx+2, ..., nx+n; ...; (n-1)n*x+1, (n-1)nx+2, ...,(n-1)nx+n].

Original entry on oeis.org

1, 6, 4, 216, 198, 36, 23040, 24640, 7200, 576, 5400000, 6375000, 2362500, 328800, 14400, 2351462400, 2982873600, 1285956000, 238533120, 19051200, 518400, 1707698764800, 2291162509440, 1100516981760, 245735819280, 27025656000, 1383117120, 25401600

Offset: 1

Vladimir Shevelev, Nov 30 2013

The degree of n-th polynomial is n-1.

Its leading coefficient is T(n,1) = n^n*(n-1)!^2*(n+1)/2. - M. F. Hasler, Dec 01 2013

			                                   1
                           6*x +   4
              216*x^2 +  198*x +  36
23040*x^3 + 24640*x^2 + 7200*x + 576
              ......

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A232773, A204248, A094638.

p[n_,x_]:=(-1)^n Sum[n^k x^k StirlingS1[n,n-k]StirlingS1[n+1,k+1](n-k)!k!,{k,0,n-1}];Flatten[Table[Reverse[CoefficientList[p[n,x],x]],{n,8}]] (* Peter J. C. Moses, Nov 30 2013 *)

P(n)=(-1)^n*sum(k=0,n-1,n^k*x^k*stirling(n,n-k)*stirling(n+1,k+1)*(n-k)!*k!)
apply(t->Vec(t),vector(7,n,P(n))) /* M. F. Hasler, Dec 01 2013 */

P_n(x) = (-1)^n * Sum_{k=0..n-1} c_k(n) * x^k, where c_k(n)= n^k * Stirling1(n,n-k) * Stirling1(n+1,k+1) * (n-k)! * k!.

P_n(1) = A232773; P_n(0) = n!^2, P_n(1/n) = A204248(n) is permanent of n X n Toeplitz matrix with the first row n,n-1,...,1 (see our comment in A204248).

More terms from Peter J. C. Moses, Nov 30 2013

A101560 Triangle read by rows giving the coefficients of general sum formulas of n-th Subfactorial numbers (A000166). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies Subf(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k).

Original entry on oeis.org

1, -2, -2, -1, 4, 11, 16, 11, 3, -10, -55, -147, -215, -179, -80, -15, 34, 305, 1247, 2910, 4224, 3904, 2245, 735, 105, -154, -1949, -10971, -35970, -76269, -109554, -108184, -72639, -31780, -8190, -945, 874, 14297, 103679, 443762, 1255671, 2484619, 3535727, 3654132, 2726787, 1434797

Offset: 1

André F. Labossière, Dec 06 2004

			Subf(7) = 7^(7 - 1) - {2 + 2*(7 - 2) + C(7 - 2,2)}*7^(7 - 2) + {4 + 11*(7 - 3) + 16*C(7 - 3,2) + 11*C(7 - 3,3) + 3*C(7 - 3,4)}*7^(7 - 3) - {10 + 55*(7 - 4) + 147*C(7 - 4,2) + 215*C(7 - 4,3)}*7^(7 - 4) + ...
= 7^6 - {2 + 10 + 10}*7^5 + {4 + 44 + 96 + 44 + 3}*7^4 - {10 + 165 + 441 + 215}*7^3 + {34 + 610 + 1247}*7^2 - {154 + 1949}*7 + {874}
= 7^6 - 22*7^5 + 191*7^4 - 831*7^3 + 1891*7^2 - 2103*7 + 874
= 117649 - 369754 + 458591 - 285033 + 92659 - 14721 + 874 = 265.

Cf. A101559, A000166, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2k} T(i,k) * C(n-k-1,i).

Original entry on oeis.org

1, 1, 3, 8, 8, 3, 9, 46, 101, 114, 65, 15, 33, 272, 975, 1935, 2289, 1615, 630, 105, 153, 1796, 9175, 26795, 49474, 60080, 48104, 24535, 7245, 945, 873, 13424, 90255, 353507, 902164, 1582455, 1953272, 1700860, 1025927, 408870, 97020, 10395, 5913

Offset: 1

André F. Labossière, Feb 01 2005

The coefficients T(i,k) along the i-th columns of the triangle are the consecutive partial sums of those found in table A094216.

			!7 = 7 + 1*C(7-2,1) + 1*C(7-2,2) + 3*C(7-3,1) + ... + 33*C(7-5,1) + 272*C(7-5,2) + 153*C(7-6,1) = 7 + 5 + 10 + 12 + 8*C(4,2) + 8*C(4,3) + 3*C(4,4) + 9*C(3,1) + 46*C(3,2) + 101*C(3,3) + 66 + 272 + 153 = 7 + 5 + 10 + 12 + 48 + 32 + 3 + 27 + 138 + 101 + 66 + 272 + 153 = 874.

Chris Zheng, Jeffrey Zheng, Triangular Numbers and Their Inherent Properties, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.

Cf. A094216, A102411, A102412, A101752, A003422, A094638, A008276.

A177977 Triangle read by rows. Polynomials based on sums of Moebius transforms.

Original entry on oeis.org

1, 1, 0, 1, 3, -2, 1, 6, 5, -6, 1, 10, 35, 26, -48, 1, 15, 85, 165, -26, -120, 1, 21, 175, 735, 1264, -36, -1440, 1, 28, 322, 1960, 5929, 8092, -1212, -10080, 1, 36, 546, 4536, 22449, 60564, 57644, -24816, -80640, 1, 45, 870, 9450, 63273, 254205, 572480

Offset: 1

Mats Granvik, May 16 2010

These polynomials were found by entering the rows of A177976 in Wolfram Alpha. The lower left half equals part of the Stirling numbers of the first kind given in table A094638. To evaluate, enter a value for n and divide row sums with factorial numbers as shown in the example section. n=-1 gives A092149, n=0 gives the Mertens function A002321, n=1 gives A000012, n=2 gives A002088, n=3 gives A015631, and n=4 gives A015634.

			Triangle begins and the polynomials are:
(1*n^0)/1
(1*n^1 +0*n^0)/1
(1*n^2 +3*n^1 -2*n^0)/2
(1*n^3 +6*n^2 +5*n^1 -6*n^0)/6
(1*n^4 +10*n^3 +35*n^2 +26*n^1 -48*n^0)/24
(1*n^5 +15*n^4 +85*n^3 +165*n^2 -26*n^1 -120*n^0)/120
(1*n^6 +21*n^5 +175*n^4 +735*n^3 +1264*n^2 -36*n^1 -1440*n^0)/720

Typo in sequence (erroneous comma) corrected by N. J. A. Sloane, May 22 2010

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).

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

Wolfdieter Lang, Oct 27 2011

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<=iA196845. 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).

			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.

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

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).

A118984 Triangular T(n,k) which contains in column k >= 0 the elements of the Stirling transform of the unsigned sequence Stirling1(j+k,j), j >= 0.

Original entry on oeis.org

1, 2, 1, 5, 6, 2, 15, 31, 23, 6, 52, 160, 195, 110, 24, 203, 856, 1505, 1365, 634, 120, 877, 4802, 11312, 14560, 10738, 4284, 720, 4140, 28337, 85225, 145096, 150325, 94444, 33228, 5040, 21147, 175896, 652703, 1404186, 1908249, 1672524, 921212, 291024

Offset: 1

Alford Arnold, May 07 2006

The initial array of unsigned Stirling numbers of the first kind (filled with an appropriate number of zeros) starts (see A094638)
     1,     0,     0,     0,     0,     0,     0,     0,  ...
     1,     1,     0,     0,     0,     0,     0,     0,  ...
     1,     3,     2,     0,     0,     0,     0,     0,  ...
     1,     6,    11,     6,     0,     0,     0,     0,  ...
     1,    10,    35,    50,    24,     0,     0,     0,  ...
     1,    15,    85,   225,   274,   120,     0,     0,  ...
     1,    21,   175,   735,  1624,  1764,   720,     0,  ...
     1,    28,   322,  1960,  6769, 13132, 13068,  5040,  ...
The Stirling transform is then applied on each individual column. - R. J. Mathar, May 19 2016.

			The array begins
     1;
     2,     1;
     5,     6,     2;
    15,    31,    23,      6;
    52,   160,   195,    110,     24;
   203,   856,  1505,   1365,    634,   120;
   877,  4802, 11312,  14560,  10738,  4284,   720;
  4140, 28337, 85225, 145096, 150325, 94444, 33228, 5040;

Sela Fried, The expected degree of noninvertibility of compositions of functions and a related combinatorial identity, arXiv:2202.13061 [math.CO], 2022. See Corollary 2.6 for a combinatorial identity of a signed version of this sequence.

Cf. A000110 (first column), A000142 (diagonal), A000670 (row sums), A003128 (2nd column), A008275, A008277.

read("transforms"):
A118984 := proc(n,k)
    [seq(0,j=0..k-2), seq( (-1)^k*combinat[stirling1](j+k,j),j=0..n)] ;
    STIRLING(%) ;
    op(n,%) ;
end proc: # R. J. Mathar, May 19 2016

Edited by R. J. Mathar, May 19 2016

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A199220 Triangle read by rows: T(n,k) = (n-1-k)*abs(s(n,n+1-k)), where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

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A199221 Triangle read by rows: T(n,k) = (n+1-k)*|s(n,n+1-k)| - 2*|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

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A218234 Infinitesimal generator for padded Pascal matrix A097805 (as lower triangular matrices).

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A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; n*x+1, n*x+2, ..., n*x+n; ...; (n-1)*n*x+1, (n-1)*n*x+2, ...,(n-1)*n*x+n].

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A101560 Triangle read by rows giving the coefficients of general sum formulas of n-th Subfactorial numbers (A000166). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies Subf(n) = Sum_{k=1..n} Sum_{i=1..2*k-1} T(i,k) * C(n-k,i-1) * n^(n-k).

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A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2*k} T(i,k) * C(n-k-1,i).

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A177977 Triangle read by rows. Polynomials based on sums of Moebius transforms.

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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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Formula

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

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A199221 Triangle read by rows: T(n,k) = (n+1-k)|s(n,n+1-k)| - 2|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; nx+1, nx+2, ..., nx+n; ...; (n-1)n*x+1, (n-1)nx+2, ...,(n-1)nx+n].

A101560 Triangle read by rows giving the coefficients of general sum formulas of n-th Subfactorial numbers (A000166). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies Subf(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k).

A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2k} T(i,k) * C(n-k-1,i).