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A190782 Triangle T(n,k), read by rows, of the coefficients of x^k in the expansion of sum_(m= 0..n) binomial(x,m)) = (a(k)*x^k)/n!, n >= 0, 0 <= k <= n. 1
1, 1, 1, 2, 1, 1, 6, 5, 0, 1, 24, 14, 11, -2, 1, 120, 94, 5, 25, -5, 1, 720, 444, 304, -75, 55, -9, 1, 5040, 3828, 364, 1099, -350, 112, -14, 1, 40320, 25584, 15980, -4340, 3969, -1064, 210, -20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

There is a strong relation between this triangle and triangle A048994 which deals with the binomial (x,n), this triangle being dealing with the summation of this binomial.

Apparently A054651 with reversed rows. - Mathew Englander, May 17 2014

LINKS

Table of n, a(n) for n=0..44.

FORMULA

T(n,k) = T(n-1,k)+ T(n-1,k-1)- T(n-2,k-1)*(n-1)+ T(n-2,k)*(n-1)^2, T(n,n)=1, T(n,0)= n! for n >= 0.

T(n,k) = T(n-1,k)*n + (A048994(n,k)), T(n,n)= 1, T(n,0)= n! for n>= 0.

EXAMPLE

Triangle begins:

n\k     0       1       2       3       4       5       6      7     8

0       1

1       1       1

2       2       1       1

3       6       5       0        1

4      24      14      11       -2      1

5     120      94       5       25     -5       1

6     720     444     304      -75     55      -9       1

7    5040    3828     364     1099   -350     112     -14      1

8   40320   25584   15980    -4340   3969   -1064     210    -20     1

...

MATHEMATICA

row[n_] := CoefficientList[ Series[ Sum[ Binomial[x, m], {m, 0, n}], {x, 0, n}], x]*n!; Table[row[n], {n, 0, 8}] // Flatten (* Jean-Fran├žois Alcover, Jan 04 2013 *)

CROSSREFS

Cf. A000142, A132393.

Sequence in context: A047920 A249673 A144655 * A330490 A199063 A140956

Adjacent sequences:  A190779 A190780 A190781 * A190783 A190784 A190785

KEYWORD

sign,tabl

AUTHOR

Mokhtar Mohamed, Dec 29 2012

STATUS

approved

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Last modified January 20 23:16 EST 2020. Contains 331104 sequences. (Running on oeis4.)