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A157744 A recursion triangle sequence: e(n,k)=Sum[(-1)^j Binomial[n + 1, j](k - j)^n, {j, 0, k}]; A(n,k)=A(n-1,k-1)+e(n-1,k). 0
1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 1, 2, 13, 17, 5, 1, 1, 2, 28, 79, 43, 6, 1, 1, 2, 59, 330, 381, 100, 7, 1, 1, 2, 122, 1250, 2746, 1572, 220, 8, 1, 1, 2, 249, 4415, 16869, 18365, 5865, 467, 9, 1, 1, 2, 504, 14857, 92649, 173059, 106599, 20473, 969, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 3, 6, 13, 38, 159, 880, 5921, 46242, 409123,...}.

The result is a nearly binomial experiment.

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 470, Equation (38).

LINKS

Table of n, a(n) for n=0..65.

FORMULA

e(n,k)=Sum[(-1)^j Binomial[n + 1, j](k - j)^n, {j, 0, k}];

A(n,k)=A(n-1,k-1)+e(n-1,k).

EXAMPLE

{1},

{1, 1},

{1, 2, 1},

{1, 2, 3, 1},

{1, 2, 6, 4, 1},

{1, 2, 13, 17, 5, 1},

{1, 2, 28, 79, 43, 6, 1},

{1, 2, 59, 330, 381, 100, 7, 1},

{1, 2, 122, 1250, 2746, 1572, 220, 8, 1},

{1, 2, 249, 4415, 16869, 18365, 5865, 467, 9, 1},

{1, 2, 504, 14857, 92649, 173059, 106599, 20473, 969, 10, 1}

MATHEMATICA

Clear[e, A, n, k];

e[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k - j)^n, {j, 0, k}];

A[1, n_] := 1;

A[n_, n_] := 1;

A[n_, k_] := A[n - 1, k - 1] + e[n - 1, k];

Table[Table[A[n, k], {k, 0, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A225641 A116855 A173265 * A030111 A096921 A308203

Adjacent sequences:  A157741 A157742 A157743 * A157745 A157746 A157747

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula and Gary W. Adamson, Mar 05 2009

STATUS

approved

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Last modified November 21 00:08 EST 2019. Contains 329348 sequences. (Running on oeis4.)