A144636 -id:A144636 - cp's OEIS frontend

A144633 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 0 <= k <= n), read by rows.

1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -5, 11, -6, 1, 0, 10, -45, 35, -10, 1, 0, 35, 175, -210, 85, -15, 1, 0, -910, -315, 1225, -700, 175, -21, 1, 0, 11935, -6265, -5670, 5565, -1890, 322, -28, 1, 0, -134750, 139755, -5005, -39270, 19425, -4410, 546, -36, 1

Offset: 0

N. J. A. Sloane, Jan 21 2009

Definition: take the triangle in A144385, write it as an (infinite) upper triangular square matrix, invert it and transpose it.

The Bell transform of A144636(n+1). Also the inverse Bell transform of the sequence "g(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
1;
0,    1;
0,   -1,    1;
0,    2,   -3,    1;
0,   -5,   11,   -6,    1;
0,   10,  -45,   35,  -10,   1;
0,   35,  175, -210,   85, -15,   1;
0, -910, -315, 1225, -700, 175, -21,  1;

J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.

Alois P. Heinz, Rows n = 0..140, flattened
J. Y. Choi and J. D. H. Smith, The Tri-restricted Numbers and Powers of Permutation Representations, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.
J. Y. Choi and J. D. H. Smith, On the Unimodality and Combinatorics of the Bessel Numbers, Discrete Math., 264 (2003), 45-53.
J. Y. Choi et al., Reciprocity for multirestricted Stirling numbers, J. Combin. Theory 113 A (2006), 1050-1060.

For another version of this triangle see A144634.
Columns give A144636-A144639.
Cf. A144402.

A:= proc(n,k) option remember; if n=k then 1 elif k A(i-1, j-1))^(-1) end:
T:= (n,k)-> M(n+1)[k+1, n+1]:
seq(seq(T(n,k), k=0..n), n=0..12); # Alois P. Heinz, Oct 23 2009

max = 10; 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; Table[A144633[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A144636(n+1), 10) # Peter Luschny, Jan 18 2016

Corrected and extended by Alois P. Heinz, Oct 23 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

sign

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

sign

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

sign

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

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A144633 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 0 <= k <= n), read by rows.

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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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