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A212857
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Number of 4 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.
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10
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1, 1, 15, 1135, 271375, 158408751, 191740223841, 429966316953825, 1644839120884915215, 10079117505143103766735, 94135092186827772028779265, 1287215725538576868883610346465, 24929029117106417518788960414909025, 664978827664071363541997348802227351425
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OFFSET
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0,3
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COMMENTS
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We generalize Daniel Suteu's recurrence from A212856. Notice first that, in the notation of Abramson and Promislow (1978), we have a(n) = R(m=4, n, t=0).
Letting y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978), we get 1 + Sum_{n >= 1} R(m,n,t=0)*x^n/(n!)^m = 1/f(-x), where f(x) = Sum_{i >= 0} (x^i/(i!)^m). Matching coefficients, we get Sum_{s = 1..n} R(m, s, t=0) * (-1)^(s-1) * binomial(n,s)^m = 1, from which the recurrence in the Formula section follows.
(End)
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LINKS
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FORMULA
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a(n) = (-1)^(n-1) + Sum_{s = 1..n-1} a(s) * (-1)^(n-s-1) * binomial(n,s)^m for n >= 2 with a(1) = 1. Here m = 4. - Petros Hadjicostas, Sep 08 2019
a(n) = (n!)^4 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^4). (see Petros Hadjicostas's comment on Sep 08 2019) - Seiichi Manyama, Jul 18 2020
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EXAMPLE
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Some solutions for n=3:
1 2 0 1 0 2 1 0 2 2 1 0 2 0 1 2 1 0 1 0 2
2 1 0 1 0 2 0 2 1 0 2 1 2 1 0 1 0 2 2 1 0
1 2 0 2 1 0 1 0 2 0 1 2 2 1 0 2 1 0 1 2 0
2 1 0 0 1 2 2 1 0 2 1 0 1 0 2 2 1 0 2 1 0
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MAPLE
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A212857 := proc(n) sum(z^k/k!^4, k = 0..infinity);
series(%^x, z=0, n+1): n!^4*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k - j], {j, 1, k}]];
a[n_] := T[4, n];
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CROSSREFS
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Cf. A000012, A000225, A000275, A212850, A212851, A212852, A212853, A212854, A212856, A212858, A212859, A212860, A336196.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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