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A177252
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Triangle read by rows: T(n,k) is the number of permutations of [n] having k adjacent 4-cycles (0 <= k <= floor(n/4)), i.e., having k cycles of the form (i, i+1, i+2, i+3).
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10
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1, 1, 2, 6, 23, 1, 118, 2, 714, 6, 5016, 24, 40201, 118, 1, 362163, 714, 3, 3623772, 5016, 12, 39876540, 40200, 60, 478639079, 362163, 357, 1, 6223394516, 3623772, 2508, 4, 87138394540, 39876540, 20100, 20, 1307195547720, 478639080, 181080, 120
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OFFSET
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0,3
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COMMENTS
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Row n contains 1 + floor(n/4) entries.
Sum of entries in row n = n! (A000142).
Sum_{k>=0} k*a(n,k) = (n-3)! (n >= 4).
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..floor(n/4)} (-1)^(k+j)*binomial(j,k)*(n-3j)!/j!.
G.f. of column k: (1/k!) * Sum_{j>=k} j! * x^(j+3*k) / (1+x^4)^(j+1). - Seiichi Manyama, Feb 24 2024
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EXAMPLE
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T(9,2)=3 because we have (1234)(5678)(9), (1234)(5)(6789), and (1)(2345)(6789).
Triangle starts:
1;
1;
2;
6;
23, 1;
118, 2;
714, 6;
5016, 24;
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MAPLE
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T := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*factorial(n-3*j)/factorial(j), j = 0 .. floor((1/4)*n)) end proc: for n from 0 to 15 do seq(T(n, k), k = 0 .. floor((1/4)*n)) end do; % yields sequence in triangular form
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MATHEMATICA
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T[n_, k_] := T[n, k] = Sum[(-1)^(k + j)*Binomial[j, k]*(n - 3 j)!/j!, {j, 0, n/4}];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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