This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320955 #43 Nov 15 2018 03:29:09
%S A320955 1,1,0,1,1,0,1,1,1,0,1,1,2,1,0,1,1,2,4,1,0,1,1,2,5,8,1,0,1,1,2,5,14,
%T A320955 16,1,0,1,1,2,5,15,41,32,1,0,1,1,2,5,15,51,122,64,1,0,1,1,2,5,15,52,
%U A320955 187,365,128,1,0,1,1,2,5,15,52,202,715,1094,256,1,0
%N A320955 Square array read by ascending antidiagonals: A(n, k) (n >= 0, k >= 0) = Sum_{j=0..n-1} (!j/j!)*((n - j)^k/(n - j)!) if k > 0 and 1 if k = 0. Here !n denotes the subfactorial of n.
%C A320955 Arndt and Sloane (see the link and A278984) identify the sequence to give "the number of words of length n over an alphabet of size b that are in standard order" and provide the formula Sum_{j = 1..b} Stirling_2(n, j) assuming b >= 1 and j >= 1. Compared to the array as defined here this misses the first row and the first column of our array.
%C A320955 The method used here is the special case of a general method described in A320956 applied to the function exp. For applications to other functions see the cross references.
%C A320955 A(k,n) is the number of color patterns (set partitions) for an oriented row of length n using up to k colors (subsets). Two color patterns are equivalent if the colors are permuted. For A(3,4) = 14, the six achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA; the eight chiral patterns are the four chiral pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - _Robert A. Russell_, Nov 10 2018
%H A320955 Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%F A320955 A(n, k) = (1/n!)*Sum_{j=0..n-1} A008290(n, n-j)*(n-j)^k if k > 0.
%F A320955 If one drops the special case A(n, 0) = 1 from the definition then column 0 becomes Sum_{k=0..n} (-1)^k/k! = A103816(n)/A053556(n).
%F A320955 Row n is given for k >= 1 by a_n(k), where
%F A320955 a_0(k) = 0^k/0!.
%F A320955 a_1(k) = 1^k/1!.
%F A320955 a_2(k) = (2^k)/2!.
%F A320955 a_3(k) = (3^k + 3)/3!.
%F A320955 a_4(k) = (6*2^k + 4^k + 8)/4!.
%F A320955 a_5(k) = (20*2^k + 10*3^k + 5^k + 45)/5!.
%F A320955 a_6(k) = (135*2^k + 40*3^k + 15*4^k + 6^k + 264)/6!.
%F A320955 a_7(k) = (924*2^k + 315*3^k + 70*4^k + 21*5^k + 7^k + 1855)/7!.
%F A320955 a_8(k) = (7420*2^k + 2464*3^k + 630*4^k + 112*5^k + 28*6^k + 8^k + 14832)/8!.
%F A320955 Note that the coefficients of the generating functions a_n are the recontres numbers A000240, A000387, A000449, ...
%F A320955 Rewriting the formulas with exponential generating functions for the rows we have egf(n) = Sum_{k=0..n} !k*binomial(n,k)*exp(x*(n-k)) and A(n, k) = (k!/n!)*[x^k] egf(n). In this formulation no special rule for the case k = 0 is needed.
%F A320955 The rows converge to the Bell numbers. Convergence here means that for every fixed k the terms in column k differ from A000110(k) only for finitely many indices.
%F A320955 A(n, n) are the Bell numbers A000110(n) for n >= 0.
%F A320955 Let S(n, k) = Bell(n+k+1) - A(n, k+n+1) for n >= 0 and k >= 0, then the square array S(n, k) read by descending antidiagonals equals provable the triangle A137650 and equals empirical the transpose of the array A211561.
%e A320955 Array starts:
%e A320955 n\k 0 1 2 3 4 5 6 7 8 9 ...
%e A320955 ----------------------------------------------------
%e A320955 [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007
%e A320955 [1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%e A320955 [2] 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ... A011782
%e A320955 [3] 1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, ... A124302
%e A320955 [4] 1, 1, 2, 5, 15, 51, 187, 715, 2795, 11051, ... A124303
%e A320955 [5] 1, 1, 2, 5, 15, 52, 202, 855, 3845, 18002, ... A056272
%e A320955 [6] 1, 1, 2, 5, 15, 52, 203, 876, 4111, 20648, ... A056273, ?A284727
%e A320955 [7] 1, 1, 2, 5, 15, 52, 203, 877, 4139, 21110, ...
%e A320955 [8] 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, ...
%e A320955 [9] 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
%e A320955 ----------------------------------------------------
%e A320955 Seen as a triangle given by the descending antidiagonals:
%e A320955 [0] 1
%e A320955 [1] 0, 1
%e A320955 [2] 0, 1, 1
%e A320955 [3] 0, 1, 1, 1
%e A320955 [4] 0, 1, 2, 1, 1
%e A320955 [5] 0, 1, 4, 2, 1, 1
%e A320955 [6] 0, 1, 8, 5, 2, 1, 1
%e A320955 [7] 0, 1, 16, 14, 5, 2, 1, 1
%p A320955 A := (n, k) -> if k = 0 then 1 else add(A008290(n, n-j)*(n-j)^k, j=0..n-1)/n! fi:
%p A320955 seq(lprint(seq(A(n, k), k=0..9)), n=0..9); # Prints the array row-wise.
%p A320955 seq(seq(A(n-k, k), k=0..n), n=0..11); # Gives the array as listed.
%t A320955 T[n_, 0] := 1; T[n_, k_] := Sum[(Subfactorial[j]/Factorial[j])((n - j)^k/(n - j)!), {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten
%t A320955 Table[Sum[StirlingS2[k, j], {j, 0, n-k}], {n, 0, 11}, {k, 0, n}] // Flatten (* _Robert A. Russell_, Nov 10 2018 *)
%Y A320955 Cf. A008290, A000110, A137650, A211561, A000240, A000387, A000449, A103816/A053556.
%Y A320955 Cf. A124302, A124303, A056272, A056273, A284727, A278984.
%Y A320955 Antidiagonal sums (and row sums of the triangle): A320964.
%Y A320955 Cf. this sequence (exp), A320962 (log(x+1)), A320956 (sec+tan), A320958 (arcsin), A320959 (arctanh).
%Y A320955 Cf. A320750 (unoriented), A320751 (chiral), A305749 (achiral).
%K A320955 nonn,tabl
%O A320955 0,13
%A A320955 _Peter Luschny_, Nov 05 2018