Formelsammlung Mathematik: Tabellen: Kombinatorik

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Binomialkoeffizienten

${\binom {n}{k}}$

n\k	−2	−1	0	1	2	3	4	5	6	7	8	9	10	11	12
−4	0	0	1	−4	10	−20	35	−56	84	−120	165	−220	286	−364	455
−3	0	0	1	−3	6	−10	15	−21	28	−36	45	−55	66	−78	91
−2	0	0	1	−2	3	−4	5	−6	7	−8	9	−10	11	−12	13
−1	0	0	1	−1	1	−1	1	−1	1	−1	1	−1	1	−1	1
0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
1	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0
2	0	0	1	2	1	0	0	0	0	0	0	0	0	0	0
3	0	0	1	3	3	1	0	0	0	0	0	0	0	0	0
4	0	0	1	4	6	4	1	0	0	0	0	0	0	0	0
5	0	0	1	5	10	10	5	1	0	0	0	0	0	0	0
6	0	0	1	6	15	20	15	6	1	0	0	0	0	0	0
7	0	0	1	7	21	35	35	21	7	1	0	0	0	0	0
8	0	0	1	8	28	56	70	56	28	8	1	0	0	0	0
9	0	0	1	9	36	84	126	126	84	36	9	1	0	0	0
10	0	0	1	10	45	120	210	252	210	120	45	10	1	0	0
11	0	0	1	11	55	165	330	462	462	330	165	55	11	1	0
12	0	0	1	12	66	220	495	792	924	792	495	220	66	12	1
13	0	0	1	13	78	286	715	1287	1716	1716	1287	715	286	78	13
14	0	0	1	14	91	364	1001	2002	3003	3432	3003	2002	1001	364	91

Stirling-Zahlen erster Art

${\bigg [}{\begin{matrix}n\\k\end{matrix}}{\bigg ]}$

n\k	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6	7	8	9
−6	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
−5	15	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
−4	65	10	1	0	0	0	0	0	0	0	0	0	0	0	0	0
−3	90	25	6	1	0	0	0	0	0	0	0	0	0	0	0	0
−2	31	15	7	3	1	0	0	0	0	0	0	0	0	0	0	0
−1	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	2	3	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0	6	11	6	1	0	0	0	0	0
5	0	0	0	0	0	0	0	24	50	35	10	1	0	0	0	0
6	0	0	0	0	0	0	0	120	274	225	85	15	1	0	0	0
7	0	0	0	0	0	0	0	720	1764	1624	735	175	21	1	0	0
8	0	0	0	0	0	0	0	5040	13068	13132	6769	1960	322	28	1	0
9	0	0	0	0	0	0	0	40320	109584	118124	67284	22449	4536	546	36	1

Stirling-Zahlen zweiter Art

${\bigg \{}{\begin{matrix}n\\k\end{matrix}}{\bigg \}}$

n\k	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6	7	8	9
−6	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
−5	15	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
−4	85	10	1	0	0	0	0	0	0	0	0	0	0	0	0	0
−3	225	35	6	1	0	0	0	0	0	0	0	0	0	0	0	0
−2	274	50	11	3	1	0	0	0	0	0	0	0	0	0	0	0
−1	120	24	6	2	1	1	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	1	3	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0	1	7	6	1	0	0	0	0	0
5	0	0	0	0	0	0	0	1	15	25	10	1	0	0	0	0
6	0	0	0	0	0	0	0	1	31	90	65	15	1	0	0	0
7	0	0	0	0	0	0	0	1	63	301	350	140	21	1	0	0
8	0	0	0	0	0	0	0	1	127	966	1701	1050	266	28	1	0
9	0	0	0	0	0	0	0	1	255	3025	7770	6951	2646	462	36	1

Berechnung am Computer

Funktion	Maxima	GAP
Binomialkoeffizient	`binomial(n,k)`	`Binomial(n,k)`
Stirling-Zahl erster Art	`abs(stirling1(n,k))`	`Stirling1(n,k)`
Stirling-Zahl zweiter Art	`stirling2(n,k)`	`Stirling2(n,k)`

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