|
Search: id:A155908
|
|
|
| A155908 |
|
New sum form triangle sequence: t0(n,k)=-Sum[(-1)^(k - j)* Binomial[n + 1, j](j)^n, {j, 0, k + 1}]/(n + 1); t(n,m)=t0(n,m)+t0(n,n-m). |
|
+0
1
|
|
| 1, 1, 1, 1, 6, 1, 1, 27, 27, 1, 1, 156, 262, 156, 1, 1, 1375, 2560, 2560, 1375, 1, 1, 16998, 33303, 34052, 33303, 16998, 1, 1, 262591, 576261, 546875, 546875, 576261, 262591, 1, 1, 4783992, 12054460, 11922248, 9222918, 11922248, 12054460, 4783992, 1, 1
(list; table; graph; listen)
|
|
|
OFFSET
|
0,5
|
|
|
COMMENT
|
Row sums are:
{1, 2, 8, 56, 576, 7872, 134656, 2771456, 66744320, 1842237440, 57354338304,...}.
The form is a modification of the Steve Roman sum definition of Stirling's numbers:
S(n,k)=Sum[Binomial[k,j]*(-1)^(k-j)*j^n,{j,0,k}]/k!
|
|
REFERENCES
|
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 60
|
|
FORMULA
|
t0(n,k)=-Sum[(-1)^(k - j)* Binomial[n + 1, j](j)^n, {j, 0, k + 1}]/(n + 1);
t(n,m)=t0(n,m)+t0(n,n-m).
|
|
EXAMPLE
|
{1},
{1, 1},
{1, 6, 1},
{1, 27, 27, 1},
{1, 156, 262, 156, 1},
{1, 1375, 2560, 2560, 1375, 1},
{1, 16998, 33303, 34052, 33303, 16998, 1},
{1, 262591, 576261, 546875, 546875, 576261, 262591, 1},
{1, 4783992, 12054460, 11922248, 9222918, 11922248, 12054460, 4783992, 1},
{1, 100002303, 287654382, 321830418, 211631616, 211631616, 321830418, 287654382, 100002303, 1},
{1, 2357952810, 7642932925, 9822446360, 6693837250, 4319999612, 6693837250, 9822446360, 7642932925, 2357952810, 1}
|
|
MATHEMATICA
|
Clear[t, n, k, j];
t[n_, k_] = -Sum[(-1)^(k - j)* Binomial[n + 1, j](j)^n, {j, 0, k + 1}]/(n + 1);
Table[Table[t[n, k], {k, 0, n - 1}], {n, 1, 10}];
Table[Table[If[n == 0, 1, (t[n, k] + t[n, n - k])], {k, 0, n}], {n, 0, 10}];
Flatten[%]
|
|
CROSSREFS
|
Sequence in context: A140945 A141688 A166960 this_sequence A105373 A111578 A166349
Adjacent sequences: A155905 A155906 A155907 this_sequence A155909 A155910 A155911
|
|
KEYWORD
|
nonn,tabl,uned
|
|
AUTHOR
|
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 30 2009
|
|
|
Search completed in 0.002 seconds
|
