A144633 -id:A144633 - cp's OEIS frontend

A144636 Column 1 of triangle in A144633.

0, 1, -1, 2, -5, 10, 35, -910, 11935, -134750, 1376375, -11761750, 48923875, 1209057850, -48943019125, 1218200233250, -25136059573625, 441933961719250, -5922915844470625, 18956805373756250, 2611377778731596875

Offset: 0

N. J. A. Sloane, Jan 23 2009

sign

E.g.f.: A(x)=B(x)-1/B(x)-1 where B(x)=(2+3*x+(5+12*x+9*x^2)^(1/2))^(1/3). [From Vladeta Jovovic, Jan 24 2009]

More terms from Vladeta Jovovic, Jan 24 2009

A144639 Column 4 of triangle in A144633.

Original entry on oeis.org

0, 0, 0, 0, 1, -10, 85, -700, 5565, -39270, 163625, 2002000, -80455375, 1796144350, -33225267075, 532997965500, -6863056074875, 39757008541250, 1589961504756625, -87655679826715000, 2971557080832965625, -82299265240798856250, 1913986621864144953125

Offset: 0

N. J. A. Sloane, Jan 23 2009

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Alois P. Heinz, Table of n, a(n) for n = 0..100

A:= proc(n, k) option remember; if n=k then 1 elif k A(i-1, j-1))^(-1) end: a:= n-> M(n+4)[5, n+1]: seq(a(n), n=0..25);  # Alois P. Heinz, Oct 25 2009

max = 30; t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; (0 <= k <= 3*n) := t[n, k] = t[n-1, k-1]+(k-1)*t[n-1, k-2]+(1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[, ] = 0; A144633 = Table[t[n, k], {n, 0, max}, {k, 0, max}] // Inverse // Transpose; a[n_] := A144633[[n+1, 5]]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Mar 19 2014 *)

E.g.f.: B(x)^4/24 where B(x) is e.g.f. for A144636. - Vladeta Jovovic, Jan 24 2009

More terms from Alois P. Heinz, Oct 25 2009

A144637 Column 2 of triangle in A144633.

Original entry on oeis.org

0, 0, 1, -3, 11, -45, 175, -315, -6265, 139755, -2127125, 28153125, -327452125, 2921393475, -2998820825, -788463550875, 28464750689375, -739460838241125, 16173782241491875, -294957734383186875, 3672958879661946875, 19544468129733421875, -3601613031568517590625

Offset: 0

N. J. A. Sloane, Jan 23 2009

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Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A144633.

A:= proc(n, k) option remember; if n=k then 1 elif k A(i-1, j-1))^(-1) end: a:= n-> M (n+2)[3, n+1]: seq (a(n), n=0..25); # Alois P. Heinz, Oct 25 2009

max = 22; t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; (0 <= k <= 3*n) := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[, ] = 0; A144633 = Table[t[n, k], {n, 0, max}, {k, 0, max}] // Inverse // Transpose ; A144633[[All, 3]] (* Jean-François Alcover, Jan 17 2014 *)

E.g.f.: B(x)^2/2 where B(x) is e.g.f. for A144636. [Vladeta Jovovic, Jan 24 2009]

More terms from Alois P. Heinz, Oct 25 2009

A144638 Column 3 of triangle in A144633.

Original entry on oeis.org

0, 0, 0, 1, -6, 35, -210, 1225, -5670, -5005, 750750, -16391375, 278528250, -4130451325, 51941839950, -429736682375, -3328691115750, 297526876521875, -10250866397021250, 276000929009430625, -6284336658393543750, 114982352685697371875, -1137436827419674031250

Offset: 0

N. J. A. Sloane, Jan 23 2009

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Alois P. Heinz, Table of n, a(n) for n = 0..100

A:= proc(n, k) option remember; if n=k then 1 elif k A(i-1, j-1))^(-1) end: a:= n-> M (n+3)[4, n+1]: seq (a(n), n=0..25);  # Alois P. Heinz, Oct 25 2009

max = 22; t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; (0 <= k <= 3*n) := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[, ] = 0; A144633 = Table[t[n, k], {n, 0, max}, {k, 0, max}] // Inverse // Transpose ; A144633[[All, 4]] (* Jean-François Alcover, Jan 17 2014 *)

E.g.f.: B(x)^3/6 where B(x) is e.g.f. for A144636. [From Vladeta Jovovic, Jan 24 2009]

More terms from Alois P. Heinz, Oct 25 2009

A144634 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 1 <= k <= n), read by rows.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -5, 11, -6, 1, 10, -45, 35, -10, 1, 35, 175, -210, 85, -15, 1, -910, -315, 1225, -700, 175, -21, 1, 11935, -6265, -5670, 5565, -1890, 322, -28, 1, -134750, 139755, -5005, -39270, 19425, -4410, 546, -36, 1, 1376375, -2127125

Offset: 1

N. J. A. Sloane, Jan 21 2009

See A144633 for further information.

			Triangle begins:
1
-1 1
2 -3 1
-5 11 -6 1
10 -45 35 -10 1
35 175 -210 85 -15 1
-910 -315 1225 -700 175 -21 1

Cf. A144633.

More terms from Vladeta Jovovic, Jan 21 2009

A171998 In general, let A(n,k,m) denote the (n,k)-th entry of the inverse of the matrix consisting of the (n,k)-th m-restrained Stirling numbers of the second kind (-1)^(n-k) times the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, as the (n+1,k+1)-th entry. The sequence shows A(n,k,3), which is a lower triangular matrix, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, -5, 7, 6, 1, -65, -15, 25, 10, 1, -455, -455, 0, 65, 15, 1, -1295, -4725, -1715, 140, 140, 21, 1

Offset: 1

Ji Young Choi, Jan 21 2010

A(n,k,m) also can be expanded for nonpositive integers n and k using the m-restrained Stirling numbers of the first kind.

			A(1,1,3) = 1, A(1,2,3) = 0, A(1,3,3) = 0, A(1,4,3) = 0, ...
A(2,1,3) = 1, A(2,2,3) = 1, A(2,3,3) = 0, A(2,4,3) = 0, ...
A(3,1,3) = 1, A(3,2,3) = 3, A(3,3,3) = 1, A(3,4,3) = 0, ...
A(4,1,3) = -5, A(4,2,3) = 7, A(4,3,3) = 6, A(4,4,3) = 1, ...
In other words, A(n,k,3) is the matrix
1
1 1
1 3 1
-5 7 6 1
...
with all other entries in each row being 0. - _N. J. A. Sloane_, Dec 21 2019

Ji Young Choi, Multi-restrained Stirling numbers, Ars Comb. 120 (2015), 113-127.
John Engbers, David Galvin, and Cliff Smyth, Restricted Stirling and Lah number matrices and their inverses, Journal of Combinatorial Theory, Series A, 161 (2019), 271-298.

Cf. A111246, A144633.

A(n,k,m) = A(n-1,k-1,m) - Sum_{i=1..m-1} (-1)^{i}(k)...(k+i-1) A(n,k+i,m) A(n,k,m) = A(n-1,k-1,m) + k A(n-1,k,m) + (-1)^m k(k+1)...(k+m-1)A(n,k+m,m).

A171996 A(n,k,m) is the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, called the (n,k)-th m-restrained Stirling numbers of the first kind, and denoted by mS_1(n,k). The sequence shows the case of m=3.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, 0, 11, -6, 1, 0, -20, 35, -10, 1, 0, 40, -135, 85, -15, 1, 0, 0, 490, -525, 175, -21, 1, 0, 0, -1120, 2905, -1540, 322, -28, 1, 0, 0, 2240, -12600, 11865, -3780, 546, -36, 1, 0, 0, 0, 47600, -76545, 38325, -8190, 870, -45, 1

Offset: 1

Ji Young Choi, Jan 21 2010

A(n,k,m) is also the (n,k)-th entry in the matrix inverting the matrix consisting of the m-restrained Stirling numbers of the second kind.

Also the Bell transform of the sequence "g(n) = [1,-1,2][n] if n < 3, otherwise 0" (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			A(1,1,3)=1,  A(1,2,3)=0,   A(1,3,3)=0,   A(1,4,3)=0, ...
A(2,1,3)=-1, A(2,2,3)=1,   A(2,3,3)=0,   A(2,4,3)=0, ...
A(3,1,3)=2,  A(3,2,3)=-3,  A(3,3,3)=1,   A(3,4,3)=0, ...
A(4,1,3)=0,  A(4,2,3)=11,  A(4,3,3)=-6,  A(4,4,3)=1, ...

J. Y. Choi, Multi-restrained Stirling numbers

Cf. A111246, A144633, A171998 (matrix inverse).

T(n,k) = (-1)^(n-k)*n!*sum(j=0, k, binomial(j,n-k-j)*binomial(k,j)*3^(-n+k+j)*2^(n-k-2*j)/k!); \\ Jinyuan Wang, Dec 21 2019

# uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: [1,-1,2][n] if n < 3 else 0, 12) # Peter Luschny, Jan 19 2016

A(n,k,m) = Sum {(-1)^(n-k)*n!} /{1^{k_1}*2^{k_2}*...*m^{k_m}*(k_1!)*(k_2!)*...*(k_m!)}, where k_1 + 2*k_2 + ... + m*k_m = n and k_1 + k_2 + ... + k_m = k.
Recurrence A(n,k,m) = Sum_{i=1..m} (-1)^(i-1)*[n-1]_{i-1}*A(n-i,k-1,m).
A(n,k,m) = A(n-1,k-1,m) - (n-1)*A(n-1,k,m) - (-1)^m(n-1)*(n-2)*...*(n-m)*A(n-m-1,k-1,m).
Generating function f(t) = (1 + t - t^2/2 + t^3/3 + ... + (-1)^(m-1) t^m/m)^x, for an indeterminate x ===> the n-th derivative of f(t) at t=0, f^(n)(0) = Sum_{k=1..n} A(n,k,m)[x]_k, where [x]_k is the k-th falling factorial
T(n,k) = (-1)^(n-k)*n!*Sum_{j=0..k} C(j,n-k-j)*C(k,j)*3^(-n+k+j)*2^(n-k-2*j)/k!. - Vladimir Kruchinin, Oct 02 2019

More terms from Peter Luschny, Jan 19 2016

cp's OEIS Frontend

A144636 Column 1 of triangle in A144633.

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A144639 Column 4 of triangle in A144633.

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A144637 Column 2 of triangle in A144633.

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A144638 Column 3 of triangle in A144633.

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A144634 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 1 <= k <= n), read by rows.

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A171996 A(n,k,m) is the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, called the (n,k)-th m-restrained Stirling numbers of the first kind, and denoted by mS_1(n,k). The sequence shows the case of m=3.

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