A174584
Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1) n X n matrices A<=J_n-I-P-P^2-P^3 with exactly two 1's in every row and column.
Original entry on oeis.org
0, 1, 31, 3114, 381022
Offset: 5
- V. S. Shevelev, Development of the rook technique for calculating the cyclic indicators of (0,1)-matrices, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 21-28 (in Russian).
- S. E. Grigorchuk, V. S. Shevelev, An algorithm of computing the cyclic indicator of couples discordant permutations with restricted position, Izvestia Vuzov of the North-Caucasus region, Nature sciences 3 (1997), 5-13 (in Russian).
A174585
Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1,2) n X n matrices A<=2(J_n-I-P-P^2-P^3) with exactly one 1 and one 2 in every row and column.
Original entry on oeis.org
0, 2, 132, 9800, 1309928
Offset: 5
- V. S. Shevelev, Development of the rook technique for calculating the cyclic indicators of (0,1)-matrices, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 21-28 (in Russian).
- S. E. Grigorchuk, V. S. Shevelev, An algorithm of computing the cyclic indicator of couples discordant permutations with restricted position, Izvestia Vuzov of the North-Caucasus region, Nature sciences 3 (1997), 5-13 (in Russian).
A383280
a(n) = (3/2)^n * Sum_{k=0..n} (1/6)^k * (2*k)! * (n-k)! * binomial(n,k)^2.
Original entry on oeis.org
1, 2, 9, 72, 954, 19980, 624510, 27420120, 1607036760, 120942324720, 11351106055800, 1298791163577600, 177888712528573200, 28728740092874421600, 5401708378739722249200, 1169716267087957140552000, 288993599402729842084464000, 80796133625685147464322528000
Offset: 0
-
a(n) = (3/2)^n*sum(k=0, n, (2*k)!*(n-k)!*binomial(n, k)^2/6^k);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(serlaplace(1/sqrt(1-x) * exp(3*x/2)))) \\ Joerg Arndt, Apr 22 2025