A113143 Table T(n,k), n >= 0 and k >= 0, read by antidiagonals, related to A111146.

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 8, 1, 1, 2, 6, 15, 16, 1, 1, 2, 7, 26, 54, 32, 1, 1, 2, 8, 41, 158, 235, 64, 1, 1, 2, 9, 60, 364, 1282, 1237, 128, 1, 1, 2, 10, 83, 708, 4409, 13158, 7790, 256, 1, 1, 2, 11, 110, 1226, 11428, 67563, 163354

Offset

0,6

Comments

Let R(m,n,k), 0 <= k <= n, the Riordan array (1, x*g(x)) where g(x)is g.f. of the m-fold factorials.

Then the row sums of R(m,n,k) are given by row m; example: m = 1, R(1,n,k) = A084938(n,k) and A051295 gives the row sums of A084938.

Square array of INVERT of m-fold factorials.

Links

Table of n, a(n) for n=0..63.

Formula

T(n, k) = Sum_{j=0..k} n^(k-j)*A111146(k, j).

Example

Table begins:
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...
1, 1, 2, 5, 15, 54, 235, 1237, 7790, 57581, 489231, 4690254, ...
1, 1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, ...
1, 1, 2, 7, 41, 364, 4409, 67573, 1248626, 26948347, 664414997, ...
1, 1, 2, 8, 60, 708, 11428, 232756, 5704964, 163192820, 5331728964, ...
1, 1, 2, 9, 83, 1226, 24727, 627909, 19169758, 682800001, 27776711627, ...
1, 1, 2, 10, 110, 1954, 47270, 1437562, 52531310, 2239259266, 109021857446, ...
1, 1, 2, 11, 141, 2928, 82597, 2925973, 124502114, 6179425823, 350316271761, ...
1, 1, 2, 12, 176, 4184, 134824, 5451528, 264710536, 14992543432, 969925065992, ...
1, 1, 2, 13, 215, 5758, 208643, 9481141, 517310894, 32922122485, 2393313188039, ...
1, 1, 2, 14, 258, 7686, 309322, 15604654, 945111938, 66766075046, 5387893860042, ...

Prog

(PARI) {T(n, k)=local(x=X+X*O(X^k), y=Y+Y*O(Y^k)); A=1/(1-x*y*sum(j=0, k, x^j*prod(i= 0, j-1, y+i))); return(sum(m=0, k, n^(k-m)*polcoeff(polcoeff(A, k, X), m, Y)))}

Crossrefs

Cf. A051295 (row n=1), A112934 (row n=2), A113144 (row n = 3), A113145 (row n=4), A113146 (row n=5), A113147 (row n = 6), A113148 (row n=7), A113149 (row n=8).

Sequence in context: A137940 A274859 A137855 * A181802 A110971 A136788

Adjacent sequences: A113140 A113141 A113142 * A113144 A113145 A113146

Keyword

nonn,tabl

Author

Philippe Deléham and Paul D. Hanna, Oct 30 2005

Status

approved