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A113143
Table T(n,k), n >= 0 and k >= 0, read by antidiagonals, related to A111146.
6
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.
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 A371823 A373778
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved