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 A277504 Array read by descending antidiagonals: T(n,k) is the number of unoriented strings with n beads of k or fewer colors. 33
 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 6, 1, 0, 1, 5, 10, 18, 10, 1, 0, 1, 6, 15, 40, 45, 20, 1, 0, 1, 7, 21, 75, 136, 135, 36, 1, 0, 1, 8, 28, 126, 325, 544, 378, 72, 1, 0, 1, 9, 36, 196, 666, 1625, 2080, 1134, 136, 1, 0, 1, 10, 45, 288, 1225, 3996, 7875, 8320, 3321, 272, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS From Petros Hadjicostas, Jul 07 2018: (Start) Column k of this array is the "BIK" (reversible, indistinct, unlabeled) transform of k,0,0,0,.... Consider the input sequence (c_k(n): n >= 1) with g.f. C_k(x) = Sum_{n>=1} c_k(n)*x^n. Let a_k(n) = BIK(c_k(n): n >= 1) be the output sequence under Bower's BIK transform. It can proved that the g.f. of BIK(c_k(n): n >= 1) is A_k(x) = (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))). (See the comments for sequence A001224.) For column k of this two-dimensional array, the input sequence is defined by c_k(1) = k and c_k(n) = 0 for n >= 1. Thus, C_k(x) = k*x, and hence the g.f. of column k is (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))) = (1/2)*(k*x/(1-k*x) + (k*x^2 + k*x)/(1-k*x^2)) = (2 + (1-k)*x - 2*k*x^2)*k*x/(2*(1-k*x^2)*(1-k*x)). Using the first form the g.f. above and the expansion 1/(1-y) = 1 + y + y^2 + ..., we can easily prove J.-F. Alcover's formula T(n,k) = (k^n + k^((n + mod(n,2))/2))/2. (End) REFERENCES See A005418. LINKS Robert A. Russell, Antidiagonals n=0..52, flattened (antidiagonals 1..50 from Andrew Howroyd) C. G. Bower, Transforms (2) FORMULA T(n,k) = [n==0] + [n>0] * (k^n + k^ceiling(n/2)) / 2. [Adapted to T(0,k)=1 by Robert A. Russell, Nov 13 2018] G.f. for column k: (1 - binomial(k+1,2)*x^2) / ((1-k*x)*(1-k*x^2)). - Petros Hadjicostas, Jul 07 2018 [Adapted to T(0,k)=1 by Robert A. Russell, Nov 13 2018] From Robert A. Russell, Nov 13 2018: (Start) T(n,k) = (A003992(k,n) + A321391(n,k)) / 2. T(n,k) = A003992(k,n) - A293500(n,k) = A293500(n,k) + A321391(n,k). G.f. for row n: (Sum_{j=0..n} S2(n,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=0..ceiling(n/2)} S2(ceiling(n/2),j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277. G.f. for row n>0: x*Sum_{k=0..n-1} A145882(n,k) * x^k / (1-x)^(n+1). E.g.f. for row n: (Sum_{k=0..n} S2(n,k)*x^k + Sum_{k=0..ceiling(n/2)} S2(ceiling(n/2),k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277. T(0,k) = 1; T(1,k) = k; T(2,k) = binomial(k+1,2); for n>2, T(n,k) = k*(T(n-3,k)+T(n-2,k)-k*T(n-1,k)). For k>n, T(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * T(n,k-j). (End) EXAMPLE Array begins with T(0,0): 1 1 1 1 1 1 1 1 1 1 ... 0 1 2 3 4 5 6 7 8 9 ... 0 1 3 6 10 15 21 28 36 45 ... 0 1 6 18 40 75 126 196 288 405 ... 0 1 10 45 136 325 666 1225 2080 3321 ... 0 1 20 135 544 1625 3996 8575 16640 29889 ... 0 1 36 378 2080 7875 23436 58996 131328 266085 ... 0 1 72 1134 8320 39375 140616 412972 1050624 2394765 ... 0 1 136 3321 32896 195625 840456 2883601 8390656 21526641 ... 0 1 272 9963 131584 978125 5042736 20185207 67125248 193739769 ... 0 1 528 29646 524800 4884375 30236976 141246028 536887296 1743421725 ... ... MATHEMATICA Table[If[n>0, ((n-k)^k + (n-k)^Ceiling[k/2])/2, 1], {n, 0, 15}, {k, 0, n}] // Flatten (* updated Jul 10 2018 *) (* Adapted to T(0, k)=1 by Robert A. Russell, Nov 13 2018 *) PROG (PARI) for(n=0, 15, for(k=0, n, print1(if(n==0, 1, ((n-k)^k + (n-k)^ceil(k/2))/2), ", "))) \\ G. C. Greubel, Nov 15 2018 (PARI) T(n, k) = {(k^n + k^ceil(n/2)) / 2} \\ Andrew Howroyd, Sep 13 2019 (Magma) [[n le 0 select 1 else ((n-k)^k + (n-k)^Ceiling(k/2))/2: k in [0..n]]: n in [0..15]]; // G. C. Greubel, Nov 15 2018 CROSSREFS Columns 0-6 are A000007, A000012, A005418(n+1), A032120, A032121, A032122, A056308. Rows 0-20 are A000012, A001477, A000217 (triangular numbers), A002411 (pentagonal pyramidal numbers), A037270, A168178, A071232, A168194, A071231, A168372, A071236, A168627, A071235, A168663, A168664, A170779, A170780, A170790, A170791, A170801, A170802. Main diagonal is A275549. Transpose is A284979. Cf. A284871, A284949. Cf. A003992 (oriented), A293500 (chiral), A321391 (achiral). Sequence in context: A339649 A349841 A339779 * A167763 A277666 A274581 Adjacent sequences: A277501 A277502 A277503 * A277505 A277506 A277507 KEYWORD nonn,tabl,easy AUTHOR Jean-François Alcover, Oct 18 2016 EXTENSIONS Array transposed for greater consistency by Andrew Howroyd, Apr 04 2017 Origin changed to T(0,0) by Robert A. Russell, Nov 13 2018 STATUS approved

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Last modified September 26 04:29 EDT 2023. Contains 365653 sequences. (Running on oeis4.)