TAB. 1: Associated k-Stirling number triangles of the second kind

$\bf {s2(k)}$, $\bf {k\neq 1}$ $\bf {s2(1)}:={\bf 1}$

k	A-number of	A-number of	A-number of
	triangle	sequence of first column	sequence of row sums
$\vdots$
-5	A049224	A025751 (Gerard)	A025759 (Gerard)
-4	A049223	A025750 (Gerard)	A025758 (Gerard)
-3	A049213	A025749(Gerard)	A025757 (Gerard)
-2	A048966	A025748 (Gerard)	A025756 (Gerard)
-1	A033184(Catalan)	A000108(n-1)	A000108(Catalan)
0	A023531 ($\bf 1$ matrix)	A000007(n-1)	A000012 (powers of 1)
2	A007318(n-1,m-1) (Pascal)	A000012	A000079(powers of 2)
3	A035324	A001700(n-1)	A049028
4	A035529	A034171(n-1)	A048965
5	A048882	A034255(n-1)	A048965
6	A049375	A034687	A039746
$\vdots$

TAB. 2: k-Stirling number triangles of the second kind

${\bf S2(k), k=0,1,2, ...,}$ ${\bf S2p(k), k=0,-1,-2,... }$

k	A-number of	A-number of	A-number of
	triangle	sequence of first column	sequence of row sums
$\vdots$
-5	A013988	A008543(n-1) (Keane)	A028844
-4	A011801	A008546(n-1) (Keane)	A028575
-3	A000369	A008545(n-1) (Keane)	A016036
-2	A004747	A008544(n-1) (Keane)	A015735
-1	A001497(n-1,m-1) (Bessel)	A001147(n-1) (double factorials)	A001515 (Riordan)
0	A023531 ($\bf 1$ matrix)	A000007(n-1)	A000012 (powers of 1)
1	A008277 (Stirling 2nd kind)	A000012 (powers of 1	A000110 (Bell)
2	A008297 (unsigned Lah)	A000142 (factorials)	A000262 (Riordan)
3	A035342	A001147 (2-factorials)	A049118
4	A035469	A007559 (3-factorials)	A049119
5	A049029	A007696 (4-factorials)	A049120
6	A049385	A008548 (5-factorials)	A049412
$\vdots$

TAB. 3: Associated k-Stirling number triangles of the first kind

${\bf s1(k)}$, $\bf {k\neq 1}$ ${\bf s1(1):= \bf 1}$

k	A-number of	A-number of	A-number of
	triangle	sequence of first column	sequence of row sums
$\vdots$
-5	A049327	A049323(5,m)	A049351
-4	A049326	A049323(4,m)	A049350
-3	A049325	A049323(3,m)	A049349
-2	A049324	A049323(2,m)	A049348
-1	A030528	A019590=A049323(1,m)	A000045(n+1) (Fibonacci)
0	A023531 ($\bf 1$ matrix)	A000007(n-1)=A049323(0,m)	A000012 (powers of 1)
2	A007318(n-1,m-1) (Pascal)	A000012 (powers of 1)	A000079 (powers of 2)
3	A030523	A001792	A039717
4	A030524	A036068	A043553
5	A030526	A036070	A045624
6	A030527	A036083	A046088
$\vdots$

TAB. 4: k-Stirling number triangles of the first kind

${\bf S1p(k), k=0,1,2, ...,}$ ${\bf S1(k), k=0,-1,-2,... }$

k	A-number of	A-number of	A-number of
	triangle	sequence of first column	sequence of row sums
$\vdots$
-5	A049411	A008279(5,n-1) (numbperm)	A049431
-4	A049424	A008279(4,n-1) (numbperm)	A049427
-3	A049410	A008279(3,n-1) (numbperm)	A049426
-2	A049404	A008279(2,n-1) (numbperm)	A049425
-1	A049403	A008279(1,n-1) (numbperm)	A000085
0	A023531 ($\bf 1$ matrix)	A000007(n-1)	A000012 (powers of 1)
1	A008275 (unsigned Stirling 1st kind)	A000142(n-1)	A000142 (factorials)
2	A008297 (unsigned Lah)	A000142 (factorials)	A000262 (Riordan)
3	A046089	A001710(n+1) (Mitrinovic2)	A049376
4	A049352	A001715(n+2) (Mitrinovic2)	A049377
5	A049353	A001720(n+3) (Mitrinovic2)	A049378
6	A049374	A001725(n+4) (Mitrinovic2)	A049402
$\vdots$

 

• About this document ...