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 A183134 Square array A(n,k) by antidiagonals. A(n,k) is the number of length 2n k-ary words (n,k>=0), either empty or beginning with the first character of the alphabet, that can be built by repeatedly inserting doublets into the initially empty word. 22
 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 10, 1, 0, 1, 1, 7, 29, 35, 1, 0, 1, 1, 9, 58, 181, 126, 1, 0, 1, 1, 11, 97, 523, 1181, 462, 1, 0, 1, 1, 13, 146, 1145, 4966, 7941, 1716, 1, 0, 1, 1, 15, 205, 2131, 14289, 48838, 54573, 6435, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS Column k > 2 is asymptotic to 2^(2*n) * (k-1)^(n+1) / ((k-2)^2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv preprint arXiv:1303.3481, 2013 A. Lakshminarayan, Z. Puchala, K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169, 2014 FORMULA A(n,k) = 1 if n=0, A(n,k) = k if n>0 and k<=1, and A(n,k) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*(k-1)^j else. A(n,k) = A183135(n,k) if n=0 or k<2, A(n,k) = A183135(n,k)/k else. G.f. of column k: 1/(1-k*x) if k<2, (1-1/k) * (1 + 2 / (k-2 + k * sqrt (1-(4*k-4)*x))) else. EXAMPLE A(3,2) = 10, because 10 words of length 6 beginning with the first character of the 2-letter alphabet {a, b} can be built by repeatedly inserting doublets (words with two equal letters) into the initially empty word: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba. Square array A(n,k) begins:   1,  1,   1,    1,    1,     1,  ...   0,  1,   1,    1,    1,     1,  ...   0,  1,   3,    5,    7,     9,  ...   0,  1,  10,   29,   58,    97,  ...   0,  1,  35,  181,  523,  1145,  ...   0,  1, 126, 1181, 4966, 14289,  ... MAPLE A:= proc(n, k)       local j;       if n=0  then 1     elif k<=1 then k               else add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1)/n       fi     end: seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA a[n_, k_] := If[ n == 0, 1 , If[ k <= 1, k, Sum [Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}] / n ] ]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *) CROSSREFS Rows 0-10 give: A000012, A057427, A004273, A079273(k) for k>0, A194716, A194717, A194718, A194719, A194720, A194721, A194722. Columns 0-10 give: A000007, A000012, A001700(n-1) for n>0, A194723, A194724, A194725, A194726, A194727, A194728, A194729, A194730. Main diagonal gives A248828. Coefficients of row polynomials for k>0 in k, (k+1) are given by A050166, A157491. Cf. A007318, A183135. Sequence in context: A231345 A271344 A327622 * A328747 A053382 A031253 Adjacent sequences:  A183131 A183132 A183133 * A183135 A183136 A183137 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 26 2010 STATUS approved

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Last modified November 25 09:04 EST 2020. Contains 338623 sequences. (Running on oeis4.)