A301741
a(n) = n! * [x^n] exp((n + 1)*x + x^2/2).
Original entry on oeis.org
1, 2, 10, 76, 778, 10026, 155884, 2839880, 59339004, 1399069450, 36746349496, 1064024248068, 33676500286840, 1156685567791586, 42850609041047760, 1703182952266379536, 72299420602524921616, 3264579136056004359570, 156238968782480840396704, 7900247992586138688381500
Offset: 0
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List([0..10],n->Sum([0..Int(n/2)],k->Factorial(n)*(n+1)^(n-2*k)/(2^k*Factorial(k)*Factorial(n-2*k)))); # Muniru A Asiru, Mar 26 2018
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Table[n! SeriesCoefficient[Exp[(n + 1) x + x^2/2], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - (n + 1) x + ContinuedFractionK[-k x^2, 1 - (n + 1) x, {k, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[Sum[n! (n + 1)^(n - 2 k)/(2^k k! (n - 2 k)!), {k, 0, Floor[n/2]}], {n, 0, 19}]
A336027
The least k such that i*k + 1 is a product of i different primes for i = 1..n.
Original entry on oeis.org
1, 10, 268, 7576, 652726, 913180816
Offset: 1
268 is in the sequence because 268 + 1 is prime, 2*268 + 1 is a product of 2 different primes, 3*268 + 1 is a product of 3 different primes, and 268 is the least number with such properties.
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nn := 1; for kk to 6 do
n := nn; do n := nextprime(n); f := true;
for k from 2 to kk do a := k*(n-1)+1; if not IsSquareFree(a) or nops(ifactors(a)[2]) <> k then f := false; break end if end do;
if f then nn := n-1; print(nn); break end if end do end do
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isok(k,n) = {for (i=1, n, if ((omega(i*k+1) != i) || (bigomega(i*k+1) != i), return (0));); return (1);}
a(n) = {my(k=1); while(!isok(k,n), k++); k;} \\ Michel Marcus, Jul 15 2020
A086363
Array T(m,n) read by antidiagonals: if X and Y are two (possibly empty) finite sets with m and n elements respectively and Z is the disjoint union of X and Y, then T(m,n) is the number of self-inverse partial functions f:Z ->Z which do not fix any element of Y.
Original entry on oeis.org
1, 1, 2, 2, 3, 5, 4, 6, 9, 14, 10, 14, 20, 29, 43, 26, 36, 50, 70, 99, 142, 76, 102, 138, 188, 258, 357, 499, 232, 308, 410, 548, 736, 994, 1351, 1850, 764, 996, 1304, 1714, 2262, 2998, 3992, 5343, 7193
Offset: 0
T(1,2)=6: If we let X={1}, Y={2,3}, so Z={1,2,3} and the relevant partial functions f:Z ->Z which do not fix either 2 or 3 are (-,-,-), (1,-,-), (-,3,2), (1,3,2), (2,1,-), (3,-,1). Here a partial function f:Z ->Z is displayed as (f(1),f(2),f(3)).
Array begins:
1, 1, 2, 4, 10, 26, 76, 232, 764, ...
2, 3, 6, 14, 36, 102, 308, 996, 3384, ...
5, 9, 20, 50, 138, 410, 1304, 4380, 15500, ...
14, 29, 70, 188, 548, 1714, 5684, 19880, 72808, ...
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T(m, n)={ if(m, if(n>1, T(m, n-1)+m*T(m-1, n-1)+(n-1)*T(m, n-2), A005425(m)+if(n,A005425(m-1)*m)),A000085(n))} \\ M. F. Hasler, Jan 13 2012
for(i=1,9,for(j=1,i,print1(T(j-1,i-j)","))) /* list values by antidiagonals */
Values double-checked using the given PARI/GP code by
M. F. Hasler, Jan 13 2012
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