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, 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

Offset: 0

Author

Mokhtar Mohamed, Dec 29 2012

Keywords

• sign
• tabl

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

Examples

			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
...
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened

Crossrefs

T(2*n,n) gives A347987.

Column 0-5 give A000142, A024167, A348063, A348064, A348065, A348068.

Cf. A048994, A054651, A132393.

Programs

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-François Alcover, Jan 04 2013 *)

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.

E.g.f. of column k: (log(1 + x))^k/(k! * (1 - x)). - Seiichi Manyama, Sep 26 2021

T(n, k) = Sum_{i=0..n-k} Stirling1(i+k, k)*n!/(i+k)!. - Igor Victorovich Statsenko, May 27 2024