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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, 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; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

LINKS

Table of n, a(n) for n=0..45.

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 * A105373 A296548 A201461

Adjacent sequences:  A155905 A155906 A155907 * A155909 A155910 A155911

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Jan 30 2009

STATUS

approved

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Last modified December 8 12:26 EST 2019. Contains 329862 sequences. (Running on oeis4.)