A232773
Permanent of the n X n matrix with numbers 1,2,...,n^2 in order across rows.
Original entry on oeis.org
1, 1, 10, 450, 55456, 14480700, 6878394720, 5373548250000, 6427291156586496, 11157501095973529920, 26968983444160450560000, 87808164603589940623344000, 374818412822626584819196231680, 2050842983500342507649178541536000, 14112022767608502582976078751055052800
Offset: 0
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a:= n-> (-1)^n*add(n^k*Stirling1(n, n-k)*
Stirling1(n+1, k+1)*(n-k)!*k!, k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Dec 02 2013
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Table[(-1)^n * Sum[n^k * StirlingS1[n, n-k] * StirlingS1[n+1, k+1] * (n-k)! * k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec after Max Alekseyev, Nov 30 2013 *)
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a(n) = (-1)^n * sum(k=0,n, n^k * stirling(n,n-k) * stirling(n+1,k+1) * (n-k)! * k! ) /* Max Alekseyev, Nov 30 2013 */
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from sympy.functions.combinatorial.numbers import stirling, factorial
def A232773(n): return abs(sum(n**k*stirling(n,n-k,kind=1,signed=True)*stirling(n+1,k+1,kind=1,signed=True)*factorial(n-k)*factorial(k) for k in range(n+1))) # Chai Wah Wu, Mar 25 2025
A232818
Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; n*x+1, n*x+2, ..., n*x+n; ...; (n-1)*n*x+1, (n-1)*n*x+2, ...,(n-1)*n*x+n].
Original entry on oeis.org
1, 6, 4, 216, 198, 36, 23040, 24640, 7200, 576, 5400000, 6375000, 2362500, 328800, 14400, 2351462400, 2982873600, 1285956000, 238533120, 19051200, 518400, 1707698764800, 2291162509440, 1100516981760, 245735819280, 27025656000, 1383117120, 25401600
Offset: 1
1
6*x + 4
216*x^2 + 198*x + 36
23040*x^3 + 24640*x^2 + 7200*x + 576
......
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p[n_,x_]:=(-1)^n Sum[n^k x^k StirlingS1[n,n-k]StirlingS1[n+1,k+1](n-k)!k!,{k,0,n-1}];Flatten[Table[Reverse[CoefficientList[p[n,x],x]],{n,8}]] (* Peter J. C. Moses, Nov 30 2013 *)
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P(n)=(-1)^n*sum(k=0,n-1,n^k*x^k*stirling(n,n-k)*stirling(n+1,k+1)*(n-k)!*k!)
apply(t->Vec(t),vector(7,n,P(n))) /* M. F. Hasler, Dec 01 2013 */
A357279
a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.
Original entry on oeis.org
1, 2, 43, 2610, 312081, 61825050, 18318396195, 7586241152490, 4184711271725985, 2965919152834367730, 2626408950849351178875
Offset: 0
a(2) = 43 because the hafnian of
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 43.
Cf.
A002024,
A002415 (absolute value of the coefficient of x^(n-2) in the characteristic polynomial of M(n)),
A095833 (k-th super- and subdiagonal sums of the matrix M(n)),
A204248 (permanent of M(n)).
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M[i_, j_, n_]:=Part[Part[Table[r+c-1,{r,n},{c,n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]
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tm(n) = matrix(n, n, i, j, i+j-1);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023
A226057
E.g.f. A(x) satisfies: A(x)^2 = -x*log(1-A(x)) where A(x) = Sum_{n>=1} a(n)*x^n/n!^2.
Original entry on oeis.org
1, 2, 21, 504, 21380, 1405800, 132139140, 16801276800, 2775758497344, 577868994460800, 147973478687496000, 45703277816543424000, 16753246307626306832640, 7190163806348621417679360, 3571395525388698501285792000
Offset: 1
E.g.f.: A(x) = x + 2*x^2/2!^2 + 21*x^3/3!^2 + 504*x^4/4!^2 + 21380*x^5/5!^2 +...
where
A(x)^2 = 2*x^2/2! + 6*x^3/3! + 34*x^4/4! + 280*x^5/5! + 3013*x^6/6! + 39963*x^7/7! + 629541*x^8/8! +...
and
-log(1-A(x)) = 2*x/2! + 6*x^2/3! + 34*x^3/4! + 280*x^4/5! + 3013*x^5/6! +...
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{a(n)=polcoeff(prod(k=0, 2*n-2, 1+k*x), n-1)*n!^2*(n-1)!/(2*n-1)!}
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from math import factorial, comb
from sympy.functions.combinatorial.numbers import stirling
def A226057(n): return factorial(n)*stirling(m:=(n<<1)-1,n,kind=1)//comb(m,n-1) # Chai Wah Wu, Jun 08 2025
Showing 1-4 of 4 results.
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